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#### Question 1:

Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:

(i) f(x) = x2 − 2x − 8
(ii) g(s) = 4s2 − 4s + 1
(iii) h(t) = t2 − 15
(iv) 6x2 − 3 − 7x
(v) $p\left(x\right)={x}^{2}+2\sqrt{2}x-6$
(vi) $q\left(x\right)=\sqrt{3}{x}^{2}+10x+7\sqrt{3}$
(vii)
(viii) g(x) = a(x2 + 1) − x(a2 + 1)
(ix) $h\left(s\right)=2{s}^{2}-\left(1+2\sqrt{2}\right)s+\sqrt{2}$
(x) $f\left(v\right)={v}^{2}+4\sqrt{3}v-15$
(xi) $p\left(y\right)={y}^{2}+\frac{3\sqrt{5}}{2}y-5$
(xii) $q\left(y\right)=7{y}^{2}-\frac{11}{3}y-\frac{2}{3}$

(i) We have,

f(x) = x2 − 2x − 8

f(x) = x2 + 2x − 4x − 8

f(x) = x (x + 2) − 4(x + 2)

f(x) = (x + 2) (x − 4)

The zeros of f(x) are given by

f(x) = 0

x2 − 2x − 8 = 0

(x + 2) (x − 4) = 0

x + 2 = 0

x = −2

Or

x − 4 = 0

x = 4

Thus, the zeros of f(x) = x2 − 2x − 8 are α = −2 and β = 4

Now,

and

Therefore, sum of the zeros =

Product of the zeros

= − 2 × 4

= −8

and

Therefore,

Product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(ii) Given

When have,

g(s) = 4s2 − 4s + 1

g(s) = 4s2 − 2s − 2s + 1

g(s) = 2s (2s − 1) − 1(2s − 1)

g(s) = (2s − 1) (2s − 1)

The zeros of g(s) are given by

Or

Thus, the zeros ofare

and

Now, sum of the zeros

and

Therefore, sum of the zeros =

Product of the zeros

and =

Therefore, the product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(iii) Given

We have,

The zeros of are given by

Now,

Sum of the zeros

and =

Therefore, sum of the zeros =

also,

Product of the zeros = αβ

and,

Therefore, the product of the zeros =

Hence, The relationship between the zeros and coefficient are verified.

(iv) Given

We have,

The zeros of are given by

Or

Thus, the zeros of  are  and.

Now,

Sum of the zeros = α + β

and, =

Therefore, sum of the zeros =

Product of the zeros = α × β

and, =

Product of zeros =

Hence, the relation between the zeros and its coefficient are verified.

(v) Given

We have,

The zeros of are given by

Or

Thus, The zeros of areand

Now,

Sum of the zeros = α + β

and,

Therefore, Sum of the zeros =

Product of the zeros

and

Therefore, The product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(vi) Given

We have,

The zeros of g(x) are given by

Or

Thus, the zeros of are and.

Now,

Sum of the zeros = α + β

and =

Therefore, sum of the zeros =

Product of zeros = α × β

and =

Therefore, the product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(vii) Given

The zeros of ƒ(x) are given by

Or

Thus, the zeros of are α = 1 and

Now,

Sum of zeros = α + β

And,

Therefore, sum of the zeros =

Product of the zeros = αβ

And

=

Product of zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(viii) Given

The zeros of g(x) are given by

or

Thus, the zeros of are

and

Sum of the zeros = α + β

and, =

Product of the zeros

And, =

Therefore,

Product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(ix) $h\left(s\right)=2{s}^{2}-\left(1+2\sqrt{2}\right)s+\sqrt{2}$

$h\left(s\right)=2{s}^{2}-s-2\sqrt{2}s+\sqrt{2}\phantom{\rule{0ex}{0ex}}h\left(s\right)=s\left(2s-1\right)-\sqrt{2}\left(2s-1\right)\phantom{\rule{0ex}{0ex}}h\left(s\right)=\left(2s-1\right)\left(s-\sqrt{2}\right)$

The zeros of h(s) are given by

h(s) = 0

Thus, the zeros of .

Now,

Sum of the zeros = $\alpha +\beta$
$=\frac{1}{2}+\sqrt{2}$

and

Therefore, sum of the zeros =

Product of the zeros
$=\frac{1}{2}×\sqrt{2}=\frac{1}{\sqrt{2}}$

and

Therefore,

Product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(x) $f\left(v\right)={v}^{2}+4\sqrt{3}v-15$

$f\left(v\right)={v}^{2}+5\sqrt{3}v-\sqrt{3}v-15\phantom{\rule{0ex}{0ex}}={v}^{2}-\sqrt{3}v+5\sqrt{3}v-15\phantom{\rule{0ex}{0ex}}=v\left(v-\sqrt{3}\right)+5\sqrt{3}\left(v-\sqrt{3}\right)\phantom{\rule{0ex}{0ex}}=\left(v-\sqrt{3}\right)\left(v+5\sqrt{3}\right)$

The zeros of f(v) are given by

f(v) = 0

Thus, the zeros of .

Now,

Sum of the zeros = $\alpha +\beta$
$=\sqrt{3}-5\sqrt{3}=-4\sqrt{3}$

and

Therefore, sum of the zeros =

Product of the zeros
$=\sqrt{3}×\left(-5\sqrt{3}\right)=-15$

and

Therefore,

Product of the zeros =

Hence, the relation-ship between the zeros and coefficient are verified.

(xi) $p\left(y\right)={y}^{2}+\frac{3\sqrt{5}}{2}y-5$
$p\left(y\right)=\frac{1}{2}\left(2{y}^{2}+4\sqrt{5}y-\sqrt{5}y-10\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\left[2y\left(y+2\sqrt{5}\right)-\sqrt{5}\left(y+2\sqrt{5}\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\left[\left(2y-\sqrt{5}\right)\left(y+2\sqrt{5}\right)\right]$
The zeros are given by p(y) = 0.
Thus, the zeros of .

Now,

Sum of the zeros = $\alpha +\beta$
$=\frac{\sqrt{5}}{2}-2\sqrt{5}=\frac{\sqrt{5}-4\sqrt{5}}{2}=\frac{-3\sqrt{5}}{2}$

and

Therefore, sum of the zeros =

Product of the zeros
$=\frac{\sqrt{5}}{2}×-2\sqrt{5}\phantom{\rule{0ex}{0ex}}=-5$

and

Therefore,

Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.

(xii) $q\left(y\right)=7{y}^{2}-\frac{11}{3}y-\frac{2}{3}$
$q\left(y\right)=\frac{1}{3}\left(21{y}^{2}-11y-2\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left(21{y}^{2}-14y+3y-2\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left[7y\left(3y-2\right)+1\left(3y-2\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left[\left(7y+1\right)\left(3y-2\right)\right]$

The zeros are given by q(y) = 0.

Thus, the zeros of .

Now,

Sum of the zeros = $\alpha +\beta$
$=\frac{-1}{7}+\frac{2}{3}\phantom{\rule{0ex}{0ex}}=\frac{11}{21}$

and

Therefore, sum of the zeros =

Product of the zeros
$=\frac{-1}{7}×\frac{2}{3}\phantom{\rule{0ex}{0ex}}=\frac{-2}{21}$

and

Therefore,

Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.

#### Question 2:

If α and β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate :

(i) α − β
(ii) $\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}$
(iii) $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-2\mathrm{\alpha \beta }$
(iv) α2β − αβ2
(v) α4 + β4
(vi) $\frac{1}{a\mathrm{\alpha }+b}+\frac{1}{a\mathrm{\beta }+b}$
(vii) $\frac{\mathrm{\beta }}{a\mathrm{\alpha }+b}+\frac{\mathrm{\alpha }}{a\mathrm{\beta }+b}$
(viii) $a\left(\frac{{\mathrm{\alpha }}^{2}}{\mathrm{\beta }}+\frac{{\mathrm{\beta }}^{2}}{\mathrm{\alpha }}\right)+b\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}\right)$

(i) Given α and are the zeros of the quadratic polynomial f (x)

=

We have,

Substituting and then we get,

Hence, the value of is.

(ii) Given α and are the zeros of the quadratic polynomial f (x)

=

We have,

Substituting and then we get,

By taking least common factor we get,

Hence the value of is .

Given α and are the zeros of the quadratic polynomial f (x)

=

We have,

By cross multiplication we get,

By substituting and we get ,

Hence the value of is.

Given α and are the zeros of the quadratic polynomial f (x)

=

We have,

By taking common factor we get,

By substituting and we get ,

Hence the value of is.

Given α and are the zeros of the quadratic polynomial f (x)

=

We have,

By substituting and we get ,

By taking least common factor we get

(vi) Since α and are the zeros of the quadratic polynomial

=

We have,

By substituting and we get ,

Hence, the value of is .

(vii) Since α and are the zeros of the quadratic polynomial

=

We have,

By substituting and we get ,

Hence, the value of is .

(viii) Since α and are the zeros of the quadratic polynomial

=

We have,

By substituting and we get ,

Hence, the value of is .

#### Question 3:

If α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x − 2, find the value of $\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}$.

Since and are the zeros of the quadratics polynomial

f (x)=

sum of zeros =

Product of the zeros =

We have,

By substituting and we get ,

Hence, the value of is .

#### Question 4:

If α and β are the zeros of the quadratic polynomial f(x) = x2x − 4, find the value of $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-\mathrm{\alpha \beta }$.

Since and are the zeros of the quadratic polynomials

sum of the zeros =

Product if zeros =

We have,

By substituting and we get ,

Hence, the value ofis.

#### Question 5:

If α and β are the zeros of the quadratic polynomial p(x) = 4x2 − 5x − 1, find the value of α2β + αβ2.

Since and are the zeros of the quadratic polynomials

Sum of the zeros =

Product of zeros =

We have,

By substituting and in, we get

Hence, the value of is.

#### Question 6:

If α and β are the zeros of the quadratic polynomial f(x) = x2 + x − 2, find the value of $\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}$.

Since and are the zeros of the quadratic polynomials

Sum of the zeros =

Product if zeros =

We have,

By substituting and we get ,

By substituting in we get ,

Taking square root on both sides we get

.

Hence, the value of is.

#### Question 7:

If α and β are the zeros of the quadratic polynomial f(x) = x2 − 5x + 4, find the value of $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-2\mathrm{\alpha \beta }$.

Sinceand are the zeros of the quadratic polynomial

Therefore

=

= 5

We have,

By substituting and we get ,

Taking least common factor we get ,

Hence, the value of is.

#### Question 8:

If α and β are the zeros of the quadratic polynomial f(t) = t2 − 4t + 3, find the value of α4β3 + α3β4.

Since and are the zeros of the quadratic polynomial

We have

Hence, the value of is .

#### Question 9:

If α and β are the zeros of the quadratic polynomial p(y) = 5y2 − 7y + 1, find the value of $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}$.

Since α and β are the zeros of the quadratic polynomial

We have,

By substituting and we get ,

Hence, the value of is.

#### Question 10:

If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4, find the value of $\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{a}}+2\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}\right)+3\mathrm{\alpha \beta }$.

Since α and β are the zeros of the quadratic polynomial

We have,

By substituting and we get ,

Hence, the value of is

#### Question 11:

If α and β are the zeros of the quadratic polynomial f(x) = x2px + q, prove that $\frac{{\mathrm{\alpha }}^{2}}{{\mathrm{\beta }}^{2}}+\frac{{\mathrm{\beta }}^{2}}{{\mathrm{\alpha }}^{2}}=\frac{{p}^{2}}{{q}^{2}}-\frac{4{p}^{2}}{q}+2$.

Since and are the zeros of the quadratic polynomial

= p

We have,

Hence, it is proved that is equal to .

#### Question 12:

If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.

Given and are the zeros of the quadratic polynomial

We have,

Substituting and then we get,

Hence, the value of is .

#### Question 13:

If the sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.

Let be the zeros of the polynomial.Then,

It is given that the sum of the zero of the quadratic polynomial is equal to their product then, we have

Hence, the value of k is

#### Question 14:

If one zero of the quadratic polynomial f(x) = 4x2 − 8kx − 9 is negative of the other, find the value of k.

Since and are the zeros of the quadratic polynomial

= 0

Hence, the Value of is .

#### Question 15:

If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find the quadratic polynomial whose zeros are .

Since α and β are the zeros of the quadratic polynomial

The roots are

Let S and P denote respectively the sum and product of zeros of the required polynomial. Then,

Taking least common factor we get,

Hence, the required polynomial is given by,

Hence, required equation is Where k is any non zero real number.

#### Question 16:

If α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2, find a quadratic polynomial whose zeros are .

Since are the zeros of the quadratic polynomial

The roots are and

Let S and P denote respectively the sum and the product of zero of the required polynomial . Then,

Taking least common factor then we have ,

By substituting and we get ,

By substituting and we get ,

Hence ,the required polynomial is given by

Hence, the required equation is Where k is any non zero real number.

#### Question 17:

If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α − β = 8, find a quadratic polynomial have α and β as its zeros.

Given

……(i)

……(ii)

By subtracting equation from we get

Substituting in equation we get,

Let S and P denote respectively the sum and product of zeros of the required polynomial. then,

Hence, the required polynomial if is given by

Hence, required equation is where k is any non-zeros real number.

#### Question 18:

If α and β are the zeros of the quadratic polynomial f(x) = x2p(x + 1) − c, show that (α + 1) (β + 1) = 1 − c.

Since and are the zeros of the quadratic polynomial

Then

=

=

We have to prove that

Substituting and we get,

Hence, it is shown that.

#### Question 19:

If α and β are the zeros of the quadratic polynomial f(x) = x2 − 2x + 3, find a polynomial whose roots are (i) α + 2, β + 2 (ii) .

(i) Since and are the zeros of the quadratic polynomial

Product of the zeros =

Let S and P denote respectively the sums and product of the polynomial whose zeros

Therefore the required polynomial f (x) is given by

Hence, the required equation is.

(ii) Since and are the zeros of the quadratic polynomial

Product of the zeros =

Let S and P denote respectively the sums and product of the polynomial whose zeros

By substituting and we get ,

The required polynomial f (x) is given by,

Hence, the required equation is , where k is any non zero real number .

#### Question 20:

If α and β are the zeros of the polynomial f(x) = x2 + px + q, from a polynomial whose zeros are (α + β)2 and (α − β)2.

If and are the zeros of the quadratic polynomial

Let S and P denote respectively the sums and product of the zeros of the polynomial whose zeros are and. Then,

The required polynomial of is given by

, where k is any non-zero real number.

#### Question 1:

Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in each case:

(i)
(ii) g(x) = x3 − 4x2 + 5x − 2; 2, 1, 1

We have,

So, and are the zeros of polynomial p(x)

Let and . Then

From

Taking least common factor we get,

From

From

Hence, it is verified that the numbers given along side of the cubic polynomials are their zeros and also verified the relationship between the zeros and coefficients

(ii) We have,

So 2,1and 1 are the zeros of the polynomial g(x)

Let and. Then,

From

From

From

Hence, it is verified that the numbers given along side of the cubic polynomials are their

zeros and also verified the relationship between the zeros and coefficients.

#### Question 2:

Find the cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, −1 and − 3 respectively.

If and are the zeros of a cubic polynomial f (x), then

where k is any non-zero real number.

Here,

Therefore

Hence, cubic polynomial is, where k is any non-zero real number.

#### Question 3:

If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.

Let and be the zeros of the polynomial

Therefore

Sum of the zeros =

Product of the zeros =

Substituting we get

Therefore, substituting and in ,and

Hence, the zeros of the polynomial are .

#### Question 4:

Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.

Let and be the zeros of the polynomials .Then,

Sum of the zeros =

Since is a zero of the polynomial .Therefore,

Substituting we get,

Hence, the condition for the given polynomial is .

#### Question 5:

If the zeros of the polynomial f(x) = ax3 + 3bx2 + 3cx + d are in A.P., prove that 2b3 − 3abc + a2d = 0.

Let and be the zeros of the polynomial f(x). Then,

Sum of the zeros =

Since a is a zero of the polynomial f(x).

Therefore,

Hence, it is proved that .

#### Question 6:

If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.

Let and be the zeros of the polynomial .

Then,

Sum of the zeros =

Since is a zero of the polynomial

Hence, the value of k is.

#### Question 1:

Apply division algorithm to find the quotient q(x) and remainder r(x) in dividing f(x) by g(x) in each of the following :

(i) f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 + x + 1
(ii) f(x) = 10x4 + 17x3 − 62x2 + 30x − 3, g(x) = 2x2 + 7x + 1
(iii) f(x) = 4x3 + 8x + 8x2 + 7, g(x) = 2x2x + 1
(iv) f(x) = 15x3 − 20x2 + 13x − 12, g(x) = 2 − 2x + x2

We have

Here, degree and

Degree

Therefore, quotient is of degree and the remainder is of degree less than 2

Let and

Using division algorithm, we have

Equating the co-efficients of various powers of on both sides, we get

On equating the co-efficient of

On equating the co-efficient of

Substituting

On equating the co-efficient of

Substituting and we get,

On equating the constant terms

Substituting we get,

Therefore,

Quotient

And remainder

Hence, the quotient and remainder is given by,

.

We have

Here, Degree and

Degree

Therefore, quotient is of degree and remainder is of degree less than 2

Let and

Using division algorithm, we have

Equating the co-efficients of various powers on both sides, we get

On equating the co-efficient of

On equating the co-efficient of

Substituting we get

On equating the co-efficient of

Substituting and, we get

On equating the co-efficient of

Substituting and,we get

On equating constant term, we get

Substituting c=-2, we get

Therefore, quotient

Remainder

Hence, the quotient and remainder are and .

we have

Here, Degree and

Degree

Therefore, quotient is of degree and

Remainder is of degree less than

Let and

Using division algorithm, we have

Equating the co-efficient of various Powers of on both sides, we get

On equating the co-efficient of

On equating the co-efficient of

Substituting we get

On equating the co-efficient of

Substituting and we get

On equating the constant term, we get

Substituting, we get

Therefore, quotient

Remainder

Hence, the quotient and remainder are and.

Given,

Here, Degree and

Degree

Therefore, quotient is of degree and

Remainder is of degree less than

Let and

Using division algorithm, we have

Equating the co-efficients of various powers of on both sides, we get

On equating the co-efficient of

On equating the co-efficient of

Substituting , we get

On equating the co-efficient of

Substituting and, we get

On equating constant term

Substituting , we get

Therefore, quotient

Remainder

Hence, the quotient and remainder are and.

#### Question 2:

Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm :

(i)
(ii)
(iii)

. Given

Here, degree and

Degree

Therefore, quotient is of degree

Remainder is of degree or less

Let and

Using division algorithm, we have

Equating co-efficient of various powers of t, we get

On equating the co-efficient of

On equating the co-efficient of

On equating the co-efficient of

Substituting, we get

On equating the co-efficient of

Substituting, we get

On equating constant term

Substituting, we get

Quotient

=

Remainder

Clearly,

Hence, is a factor of

(ii) Given

Here, Degree and

Degree

Therefore, quotient is of degree

Remainder is of degree1

Let and

Using division algorithm, we have

Equating the co-efficient of various powers of on both sides, we get

On equating the co-efficient of

On equating the co-efficient of

On equating the co-efficient of

Substituting we get

On equating the co-efficient of

Substituting and, we get

On equating constant term, we get

Substituting, we get

Therefore, quotient

Remainder

Clearly,

Hence, is not a factor of

(iii) Given,

Here, Degree and

Degree

Therefore, quotient is of degree and

Remainder is of degree less than

Let and

Using division algorithm, we have

Equating the co-efficient of various powers of on both sides, we get

On equating the co-efficient of

On equating the co-efficient of

Substituting, we get

On equating the co-efficient of

Substituting and, we get

On equating the co-efficient of

Substituting, we get

On equating the co-efficient of

Substituting and, we get

On equating constant term

Substituting, we get

Therefore, Quotient

Remainder

Clearly,

Hence, is a factor of

#### Question 3:

Obtain all zeros of the polynomial f(x) = 2x4 + x3 − 14x2 − 19x − 6, if two of its zeros are −2 and −1.

We know that, if is a zero of a polynomial, and then is a factor of.

Since and are zeros of .

Therefore

is a factor of .Now, We divide by to find the other zeros of .

By using division algorithm we have,

Hence, the zeros of the given polynomials are.

#### Question 4:

Obtain all zeros of f(x) = x3 + 13x2 + 32x + 20, if one of its zeros is −2.

Since −2 is one zero of

Therefore, we know that, if is a zero of a polynomial, then is a factor of is a factor of .

Now, we divide by to find the others zeros of .

By using that division algorithm we have,

Hence, the zeros of the given polynomials are .

#### Question 5:

Obtain all zeros of the polynomial f(x) = x4 − 3x3x2 + 9x − 6, if two of its zeros are

We know that if is a zero of a polynomial, then is a factor of .

Since and are zeros of .

Therefore

is a factor of .Now, we divide by to find the other zeros of .

By using division algorithm we have, .

Hence, the zeros of the given polynomials are .

#### Question 6:

Find all zeros of the polynomial f(x) = 2x4 − 2x3 − 7x2 + 3x + 6, if its two zeros are .

Since and are two zeros of .Therefore,

is a factor of .

Also is a factor of .

Let us now divide by. We have,

By using division algorithm we have,

Hence, The zeros of are .

#### Question 7:

What must be added to the polynomial f(x) = x4 + 2x3 − 2x2 + x − 1 so that the resulting polynomial is exactly divisible by x2 + 2x − 3?

We know that,

Clearly , Right hand side is divisible by .

Therefore, Left hand side is also divisible by .Thus, if we add to , then the resulting polynomial is divisible by.

Let us now find the remainder when is divided by.

Hence, we should add to so that the resulting polynomial is divisible by .

#### Question 8:

What must be subtracted from the polynomial f(x) = x4 + 2x3 − 13x2 − 12x + 21 so that the resulting polynomial is exactly divisible by x2 − 4x + 3?

We know that DividendQuotientDivisorRemainder.

DividendRemainderQuotientDivisor.

Clearly, Right hand side of the above result is divisible by the divisor.

Therefore, left hand side is also divisible by the divisor.

Thus, if we subtract remainder from the dividend, then it will be exactly divisible by the divisor.

Dividing by

Therefore, quotient and remainder.

Thus, if we subtract the remainder from , it will be divisible by .

#### Question 9:

Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120, if two of its zeros are 2 and −2.

We know that if is a zero of a polynomial, then is a factor of .

Since, and are zeros of .

Therefore

is a factor of .Now, we divide by to find the other zeros of .

By using division algorithm we have

Hence, the zeros of the given polynomial are .

#### Question 10:

Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30, if two of its zeros are .

We know that if is a zero of a polynomial, and then is a factor of .

Since and are zeros of .

Therefore

is a factor of . Now, we divide by to find the zero of .

By using division algorithm we have

Hence, the zeros of the given polynomial are .

#### Question 11:

Find all the zeros of the polynomial 2x3 + x2 − 6x − 3, if two of its zeros are .

We know that if is a zero of a polynomial, and then is a factor of .

Since and are zeros of .

Therefore

is a factor of .Now, we divide by to find the other zeros of .

By using division algorithm we have

Hence, the zeros of the given polynomial are.

#### Question 12:

Find all the zeros of the polynomial x3 + 3x2 − 2x − 6, if two of its zeros are .

We know that if is a zero of a polynomial, and then is a factor of .

Since and are zeros of .

Therefore

is a factor of .Now, we divide by to find the other zeros of .

By using division algorithm we have

Hence, the zeros of the given polynomials are .

#### Question 1:

Define a polynomial with real coefficients.

In the polynomial,

, and are known as the terms of the polynomial and and are their real coefficients.

For example, is a polynomial and 3 is a real coefficient

#### Question 2:

Define degree of a polynomial.

The exponent of the highest degree term in a polynomial is known as its degree.

In other words, the highest power of x in a polynomial is called the degree of the polynomial.

For Example: is a polynomial in the variable x of degree 2.

#### Question 3:

Write the standard form of a linear polynomial with real coefficients.

Any linear polynomial in variable with real coefficients is of the form, where are real numbers and

#### Question 4:

Write the standard form of a quadratic polynomial with real coefficients.

Any quadratic polynomial in variable with real coefficients is of the form, where are real numbers and

#### Question 5:

Write the standard form of a cubic polynomial with real coefficients.

The most general form of a cubic polynomial with coefficients as real numbers is of the form, where are real number and

#### Question 6:

Define value of polynomial at a point.

If is a polynomial and is any real number, then the real number obtained by replacing by in, is called the value of at and is denoted by

#### Question 7:

Define the zero of a polynomial.

The zero of a polynomial is defined as any real number such that

#### Question 8:

The sum and product of the zeros of a quadratic polynomial are $-\frac{1}{2}$ and −3 respectively. What is the quadratic polynomial.

Let sum of quadratic polynomial is

Product of the quadratic polynomial is

Let S and P denote the sum and product of the zeros of a polynomial asand.

Then

The required polynomial is given by

Hence, the quadratic polynomial is, where k is any non-zero real number

#### Question 9:

Write the family of quadratic polynomials having $-\frac{1}{4}$ and 1 as its zeros.

We know that, if is a zero of a polynomial then is a factor of quadratic polynomials.

Sinceand are zeros of polynomial.

Therefore

Hence, the family of quadratic polynomials is, where k is any non-zero real number

#### Question 10:

If the product of zeros of the quadratic polynomial f(x) = x2 − 4x + k is 3, find the value of k.

We have to find the value of k.

Given,

The product of the zeros of the quadratic polynomial .is

Product of the polynomial

Hence, the value of k is.

#### Question 11:

If the sum of the zeros of the quadratic polynomial f(x) = kx2 − 3x + 5 is 1, write the value of k.

We have to find the value of k, if the sum of the zeros of the quadratic polynomial is

Given

Sum of the polynomial

Hence, the value of k is

#### Question 12:

In Fig. 2.17, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.

Just see the point of intersection of the curve and x-axis and find out the x-coordinate of these points. These x-coordinates will be the zeros of the polynomial

Since the intersection points are

Hence, the zeros of the polynomial is

#### Question 13:

The graph of a polynomial y = f(x), shown in Fig. 2.18. Find the number of real zeros of f(x).

A real number is a zero of polynomial, if

In the above figure the curve intersects x-axis at one point and touches at one point

When a curve touches x-axis at one point, it means it has two common zeros at that point

Hence the number of real zeroes is

#### Question 14:

The graph of the polynomial f(x) = ax2 + bx + c is as shown below (Fig. 2.19). Write the signs of 'a' and b2 − 4ac.

Clearly, represent a parabola opening upwards. Therefore,

Since the parabola cuts x-axis at two points, this means that the polynomial will have two real solutions

Hence

Hence and

#### Question 15:

The graph of the polynomial f(x) = ax2 + bx + c is as shown in Fig. 2.20. Write the value of b2 − 4ac and the number of real zeros of f(x).

The graph of the polynomial or the curve touches x−axis at point. The x-coordinate of this point gives two equal zeros of the polynomial and.

Hence the number of real zeros of is 2 and

#### Question 16:

In Q. No. 14, write the sign of c.

The parabola cuts y-axis at point P which lies on y-axis. Putting in , we get y = c. So the coordinates of P are. Clearly, P lies on OY. Therefore

Hence, the sign of c is

#### Question 17:

In Q. No. 15, write the sign of c.

The parabola cuts y-axis at P which lies on OY.

Putting in, we get y=c. So the coordinates of P are. Clearly, P lies on. Therefore

#### Question 18:

The graph of a polynomial f(x) is as shown in Fig. 2.21. Write the number of real zeros of f(x).

The graph of a polynomial touches x−axis at two points

We know that if a curve touches the x-axis at two points then it has two common zeros of .

Hence the number of zeros of, in this case is 2.

#### Question 19:

If x = 1 is a zero of the polynomial f(x) = x3 − 2x2 + 4x + k, write the value of k.

We have to find the value of K if is a zero of the polynomial f(x) = x32x2 + 4x + k.

Hence, the value of k is

#### Question 20:

State division algorithm for polynomials.

If and are any two polynomials with then we can always find polynomials and such that, where or degree degree

#### Question 21:

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x), q(x) + r(x), where degree r(x) = 0.

Using division algorithm, we have

Hence an example for polynomial,, and satisfying are

#### Question 22:

Write a quadratic polynomial, sum of whose zeros is $2\sqrt{3}$ and their product is 2.

Let S and P denotes respectively the sum and product of the zeros of a polynomial are and.

The required polynomial g(x) is given by

Hence, the quadratic polynomial is where k is any non-zeros real number.

#### Question 23:

If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.

Here represent dividend and represent divisor.

Therefore degree of

Degree of

The quotient q(x) is of degree

The remainder is of degree or less.

Hence, the degree of the remainder is equal to or less than

#### Question 24:

If f(x) = x3 + x2ax + b is divisible by x2 x write the value of a and b.

We are given is exactly divisible by then the remainder should be zero

Therefore Quotient and

Remainder

Now, Remainder

Equating coefficient of x, we get

.

Equating constant term

Hence, the value of a and b are

#### Question 25:

If ab, a and b are zeros of the polynomial f(x) = 2x3 − 6x2 + 5x − 7, write the value of a.

Let a − b, a and a + b be the zeros of the polynomial then

Sum of the zeros =

Hence, the value of a is.

#### Question 26:

Write the coefficient of the polynomial p(z) = z5 − 2z2 + 4.

We have to find the co-efficient of the polynomial

Co-efficient of

Co-efficient of

Co-efficient of

Co-efficient of

Co-efficient of

Constant term

Hence, the co-efficient of and constant term is

#### Question 27:

Write the zeros of the polynomial x2x − 6.

We have to find the zeros of the polynomial

We know that if is a factor of then is a zero of polynomial

Therefore we have

Also

Hence, the zeros of polynomial is

#### Question 28:

If (x + a) is a factor of 2x2 + 2ax + 5x + 10, find a.

Given is a factor of.

Let us now divide by.

We have,

Now, remainder

Hence, the value of a is

#### Question 29:

For what value of k, −4 is a zero of the polynomial x2x − (2k + 2)?

We know that if is zero polynomial then is a factor of

Since is zero of

Therefore is a factor of

Now, we divide by to find the value of k

Now, Remainder

Hence, the value of k is

#### Question 30:

If 1 is a zero of the polynomial p(x) = ax2 − 3(a − 1) x − 1, then find the value of a.

We know that if is a zero of polynomial then is a factor of

Since is zero of

Therefore, is a factor of

Now, we divide by x − 1.

Now, Remainder

Hence, the value of a is

#### Question 31:

If α, β are the zeros of a polynomial such that α + β = −6 and αβ = −4, then write the polynomial.

Let S and P denotes respectively the sum and product of the zeros of a polynomial

We are given S = and P =. Then

The required polynomial is given by

Hence, the polynomial is

#### Question 32:

If α, β are the zeros of the polynomial 2y2 + 7y + 5, write the value of α + β + αβ.

Let and are the zeros of the polynomial .Then

The sum of the zeros The product of the zeros =

Then the value of is

Hence, the value of is

#### Question 33:

For what value of k, is 3 a zero of the polynomial 2x2 + x + k?

We know that if is zero polynomial, and then is a factor of

Since is zero of

Therefore is a factor of

Now, we divide by to find the value of k

Now, remainder

Hence, the value of k is

#### Question 34:

For what value of k, is −3 a zero of the polynomial x2 + 11x + k?

We know that if is zeros polynomial, then is a factor of

Since is zero of . Therefore is a factor of

Now, we divide by to find the value of k.

Now, Remainder

Hence, the value of k is.

#### Question 35:

For what value of k, is −2 a zero of the polynomial 3x2 + 4x + 2k?

We know that if is zero polynomial then is a factor of

Since is a factor of .Therefore is a factor of

Now, we divide by to find the value of k

Now, Remainder

Hence, the value of k is

#### Question 36:

If a quadratic polynomial f(x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f(x)?

If a quadratic polynomial is factorized into linear polynomials then the total number of real and distinct zeros of will be.

#### Question 37:

If a quadratic polynomial f(x) is a square of a linear polynomial, then its two zeros are coincident. (True/False).

The polynomial has two identical factors. The curve cuts X axis at two coincident points that is exactly at one point.

Hence, quadratic polynomial is a square of linear polynomial then its two zeros are coincident.

#### Question 38:

If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True/False)

When polynomial is not factorizable then the curve does not touch x-axis. Parabola open upwards above the x-axis or open downwards below x-axis where or

Hence, if quadratic polynomial is not factorizable into linear factors then it has no real zeros. .

#### Question 39:

If f(x) is a polynomial such that f(a) f(b) < 0, then what is the number of zeros lying between a and b?

If is a polynomial such that then this means the value of the polynomial are of different sign for a to b

Hence, at least one zero will be lying between a and b

#### Question 40:

If graph of quadratic polynomial ax2 + bx + c cuts positive direction of y-axis, then what is the sign of c?

If graph of quadratic polynomial cuts positive direction of y−axis, then

Put x = 0 for the point of intersection of the polynomial and y−axis

We have

Since the point is above the x-axis

Hence, the sign of c is positive, that is

#### Question 41:

If the graph of quadratic polynomial ax2 + bx + c cuts negative direction of y-axis, then what is the sign of c?

Since graph of quadratic polynomial cuts negative direction of y−axis

So put x=0 to find the intersection point on y-axis

So the point is

Now it is given that the quadratic polynomial cuts negative direction of y

So

#### Question 1:

If α, β are the zeros of the polynomial f(x) = x2 + x + 1, then $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}=$

(a) 1
(b) −1
(c) 0
(d) None of these

Since and are the zeros of the quadratic polynomial

We have

The value of is

Hence, the correct choice is .

#### Question 2:

If α, β are the zeros of the polynomial p(x) = 4x2 + 3x + 7, then $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}$ is equal to

(a) $\frac{7}{3}$
(b) $-\frac{7}{3}$
(c) $\frac{3}{7}$
(d) $-\frac{3}{7}$

Since and are the zeros of the quadratic polynomial

We have

The value of is.

Hence, the correct choice is

#### Question 3:

If one zero of the polynomial f(x) = (k2 + 4)x2 + 13x + 4k is reciprocal of the other, then k =

(a) 2
(b) −2
(c) 1
(d) −1

We are given then

One root of the polynomial is reciprocal of the other. Then, we have

1

Hence the correct choice is

#### Question 4:

If the sum of the zeros of the polynomial f(x) = 2x3 − 3kx2 + 4x − 5 is 6, then the value of k is

(a) 2
(b) 4
(c) −2
(d) −4

Let, be the zeros of the polynomial and we are given that
$\alpha +\beta +\gamma$=.

Then,

It is given that

$\alpha +\beta +\gamma$=

Substituting $\alpha +\beta +\gamma =\frac{3k}{2}$, we get

The value of k is.

Hence, the correct alternative is

#### Question 5:

If α and β are the zeros of the polynomial f(x) = x2 + px + q, then a polynomial having is its zero is

(a) x2 + qx + p
(b) x2px + q
(c) qx2 + px + 1
(d) px2 + qx + 1

Let and be the zeros of the polynomial.Then,

And

Let S and R denote respectively the sum and product of the zeros of a polynomial

Whose zeros are and .then

Hence, the required polynomial whose sum and product of zeros are S and R is given by

So

Hence, the correct choice is

#### Question 6:

If α, β are the zeros of polynomial f(x) = x2p (x + 1) − c, then (α + 1) (β + 1) =

(a) c − 1
(b) 1 − c
(c) c
(d) 1 + c

Since and are the zeros of quadratic polynomial

We have

The value of is.

Hence, the correct choice is

#### Question 7:

If α, β are the zeros of the polynomial f(x) = x2p(x + 1) − c such that (α +1) (β + 1) = 0, then c =

(a) 1
(b) 0
(c) −1
(d) 2

Since and are the zeros of quadratic polynomial

We have

The value of c is.

Hence, the correct alternative is

#### Question 8:

If f(x) = ax2 + bx + c has no real zeros and a + b + c = 0, then

(a) c = 0
(b) c > 0
(c) c < 0
(d) None of these

If has no real zeros and then

Hence, the correct choice is

#### Question 9:

If the diagram in Fig. 2.22 shows the graph of the polynomial f(x) = ax2 + bx + c, then

(a) a > 0, b < 0 and c > 0
(b) a < 0, b < 0 and c < 0
(c) a < 0, b > 0 and c > 0
(d) a < 0, b > 0 and c < 0

Clearly, represent a parabola opening upwards.

Therefore, cuts Y axis at P which lies on . Putting x = 0 in , we get y = c. So the coordinates of P is. Clearly, P lies on. Therefore

Hence, the correct choice is

#### Question 10:

Figure 2.23 show the graph of the polynomial f(x) = ax2 + bx + c for which

(a) a < 0, b > 0 and c > 0
(b) a < 0, b < 0 and c > 0
(c) a < 0, b < 0 and c < 0
(d) a > 0, b > 0 and c < 0

Clearly, represent a parabola opening downwards. Therefore,

cuts y-axis at P which lies on. Putting x = 0 in, we get y = c. So the coordinates P are. Clearly, P lies on. Therefore

The vertex of the parabola is in the second quadrant. Therefore,

Therefore and

Hence, the correct choice is

#### Question 11:

If the product of zeros of the polynomial f(x) ax3 − 6x2 + 11x − 6 is 4, then a =

(a) $\frac{3}{2}$
(b) $-\frac{3}{2}$
(c) $\frac{2}{3}$
(d) $-\frac{2}{3}$

Since and are the zeros of quadratic polynomial

So we have

The value of a is

Hence, the correct alternative is.

#### Question 12:

If zeros of the polynomial f(x) = x3 − 3px2 + qxr are in A.P., then

(a) 2p3 = pqr
(b) 2p3 = pq + r
(c) p3 = pqr
(d) None of these

Let be the zeros of the polynomial then

Since a is a zero of the polynomial

Therefore,

Substituting .we get

Hence, the correct choice is

#### Question 13:

If the product of two zeros of the polynomial f(x) = 2x3 + 6x2 − 4x + 9 is 3, then its third zero is

(a) $\frac{3}{2}$
(b) $-\frac{3}{2}$
(c) $\frac{9}{2}$
(d) $-\frac{9}{2}$

Let be the zeros of polynomial such that

We have,

Putting in, we get

Therefore, the value of third zero is.

Hence, the correct alternative is.

#### Question 14:

If the polynomial f(x) = ax3 + bxc is divisible  by the polynomial g(x) = x2 + bx + c, then ab =

(a) 1
(b) $\frac{1}{c}$
(c) −1
(d) $-\frac{1}{c}$

We have to find the value of

Given is divisible by the polynomial

We must have

, for all x

So put x = 0 in this equation

Since , so

Hence, the correct alternative is .

#### Question 15:

If Q.No. 14, c =

(a) b
(b) 2b
(c) 2b2
(d) −2b

We have to find the value of c

Given is divisible by the polynomial

We must have

for all

…… (1)

Since , so

Now in the equation (1) the condition is true for all x. So put x = 1 and also we have ab = 1

Therefore we have

Substituting and we get,

Hence, the correct alternative is

#### Question 16:

If one root of the polynomial f(x) = 5x2 + 13x + k is reciprocal of the other, then the value of k is

(a) 0
(b) 5
(c) $\frac{1}{6}$
(d) 6

If one zero of the polynomial is reciprocal of the other. So

Now we have

Since

Therefore we have

Hence, the correct choice is

#### Question 17:

If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}+\frac{1}{\mathrm{\gamma }}=$

(a) $-\frac{b}{d}$
(b) $\frac{c}{d}$
(c) $-\frac{c}{d}$
(d) $-\frac{c}{a}$

We have to find the value of

Given be the zeros of the polynomial

We know that

So

Hence, the correct choice is

#### Question 18:

If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then α2 + β2 + γ2 =

(a) $\frac{{b}^{2}-ac}{{a}^{2}}$
(b) $\frac{{b}^{2}-2ac}{a}$
(c) $\frac{{b}^{2}+2ac}{{b}^{2}}$
(d) $\frac{{b}^{2}-2ac}{{a}^{2}}$

We have to find the value of

Given be the zeros of the polynomial

Now

The value of

Hence, the correct choice is

#### Question 19:

If α, β, γ are are the zeros of the polynomial f(x) = x3px2 + qxr, then $\frac{1}{\mathrm{\alpha \beta }}+\frac{1}{\mathrm{\beta \gamma }}+\frac{1}{\mathrm{\gamma \alpha }}=$

We have to find the value of

Given be the zeros of the polynomial

Now we calculate the expression

Hence, the correct choice is

#### Question 20:

If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then $\frac{1}{{\mathrm{\alpha }}^{2}}+\frac{1}{{\mathrm{\beta }}^{2}}=$

(a) $\frac{{b}^{2}-2ac}{{a}^{2}}$
(b) $\frac{{b}^{2}-2ac}{{c}^{2}}$
(c) $\frac{{b}^{2}+2ac}{{a}^{2}}$
(d) $\frac{{b}^{2}+2ac}{{c}^{2}}$

We have to find the value of

Given and are the zeros of the quadratic polynomial f(x)

We have,

Hence, the correct choice is

#### Question 21:

If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is

(a) $\frac{-d}{a}$
(b) $\frac{c}{a}$
(c) $\frac{-b}{a}$
(d) $\frac{b}{a}$

Let and be the zeros of the polynomial

Therefore

The value of

Hence, the correct choice is

#### Question 22:

If two zeros x3 + x2 − 5x − 5 are , then its third zero is

(a) 1
(b) −1
(c) 2
(d) −2

Let and be the given zeros and be the third zero of x3 + x2 − 5x − 5 = 0 then

By substitutingand in

Hence, the correct choice is

#### Question 23:

The product of the zeros of x3 + 4x2 + x − 6 is

(a) −4
(b) 4
(c) 6
(d) −6

Given be the zeros of the polynomial

Product of the zeros =

The value of Product of the zeros is 6.

Hence, the correct choice is

#### Question 24:

What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?

(a) 1
(b) 2
(c) 4
(d) 5

If, is a zero of a polynomial thenis a factor of

Since 3 is the zero of the polynomial,

Thereforeis a factor of

Now, we divideby we get

Therefore we should add 2 to the given polynomial

Hence, the correct choice is

#### Question 25:

What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?

(a) 30
(b) 14
(c) 15
(d) 16

We know that, if, is zero of a polynomial then is a factor of

Since 15 is zero of the polynomial f (x) = x2 − 16x + 30, therefore (x − 15) is a factor of f (x)

Now, we divide by we get

Thus we should subtract the remainder from,

Hence, the correct choice is.

#### Question 26:

A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is

(a) x2 − 9
(b) x2 + 9
(c) x2 + 3
(d) x2 − 3

Since and are the zeros of the quadratic polynomials such that

If one of zero is 3 then

Substituting in we get

Let S and P denote the sum and product of the zeros of the polynomial respectively then

Hence, the required polynomials is

Hence, the correct choice is

#### Question 27:

If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3, then its third zero is

(a) −1
(b) 1
(c) −9
(d) 9

Let and be the given zeros and be the third zero of the polynomial then

Substituting and in, we get

Hence, the correct choice is.

#### Question 28:

If are two zeroes of the polynomial x3 + 3x2 − 5x − 15, then its third zero is

(a) 3
(b) −3
(c) 5
(d) −5

Let andbe the given zeros and be the third zero of the polynomial . Then,

Substitutingandin

We get

Hence, the correct choice is

#### Question 29:

If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then

(a) a= 1, b = 3
(b) a = 3, b = 1
(c) a = −1, b = 5
(d) a = 5, b = −1

Given that is a factor of and a + b=4

By solving and a + b = 4 by elimination method we get

Multiply by we get,

. So

By substituting b = 1 in a + b = 4 we get

Then a = 3, b = 1

Hence, the correct choice is

#### Question 30:

The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is

(a) x3 − 3x2 + 3x − 5
(b) −x3 − 3x2 − 3x − 5
(c) −x3 + 3x2 − 3x + 5
(d) x3 − 3x2 − 3x + 5