RD Sharma Solutions for Class 10 Math Chapter 2 Polynomials are provided here with simple step-by-step explanations. These solutions for Polynomials are extremely popular among class 10 students for Math Polynomials Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma Book of class 10 Math Chapter 2 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma Solutions. All RD Sharma Solutions for class 10 Math are prepared by experts and are 100% accurate.
Page No 2.33:
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)
(vi)
(vii)
(viii) g(x) = a(x2 + 1) − x(a2 + 1)
(ix)
(x)
(xi)
(xii)
Answer:
(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)
The zeros of h(s) are given by
h(s) = 0
Thus, the zeros of .
Now,
Sum of the zeros =
and
Therefore, sum of the zeros =
Product of the zeros
and
Therefore,
Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.
(x)
The zeros of f(v) are given by
f(v) = 0
Thus, the zeros of .
Now,
Sum of the zeros =
and
Therefore, sum of the zeros =
Product of the zeros
and
Therefore,
Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.
(xi)
The zeros are given by p(y) = 0.
Thus, the zeros of .
Now,
Sum of the zeros =
and
Therefore, sum of the zeros =
Product of the zeros
and
Therefore,
Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.
(xii)
The zeros are given by q(y) = 0.
Thus, the zeros of .
Now,
Sum of the zeros =
and
Therefore, sum of the zeros =
Product of the zeros
and
Therefore,
Product of the zeros =
Hence, the relation-ship between the zeros and coefficient are verified.
Page No 2.33:
Question 2:
If α and β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate :
(i) α − β
(ii)
(iii)
(iv) α2β − αβ2
(v) α4 + β4
(vi)
(vii)
(viii)
Answer:
(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 .
Page No 2.33:
Question 3:
If α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x − 2, find the value of .
Answer:
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 .
Page No 2.33:
Question 4:
If α and β are the zeros of the quadratic polynomial f(x) = x2 − x − 4, find the value of .
Answer:
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.
Page No 2.33:
Question 5:
If α and β are the zeros of the quadratic polynomial p(x) = 4x2 − 5x − 1, find the value of α2β + αβ2.
Answer:
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.
Page No 2.34:
Question 6:
If α and β are the zeros of the quadratic polynomial f(x) = x2 + x − 2, find the value of .
Answer:
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.
Page No 2.34:
Question 7:
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 5x + 4, find the value of .
Answer:
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.
Page No 2.34:
Question 8:
If α and β are the zeros of the quadratic polynomial f(t) = t2 − 4t + 3, find the value of α4β3 + α3β4.
Answer:
Since and are the zeros of the quadratic polynomial
We have
Hence, the value of is .
Page No 2.34:
Question 9:
If α and β are the zeros of the quadratic polynomial p(y) = 5y2 − 7y + 1, find the value of .
Answer:
Since α and β are the zeros of the quadratic polynomial
We have,
By substituting and we get ,
Hence, the value of is.
Page No 2.34:
Question 10:
If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4, find the value of .
Answer:
Since α and β are the zeros of the quadratic polynomial
We have,
By substituting and we get ,
Hence, the value of is
Page No 2.34:
Question 11:
If α and β are the zeros of the quadratic polynomial f(x) = x2 − px + q, prove that .
Answer:
Since and are the zeros of the quadratic polynomial
= p
We have,
Hence, it is proved that is equal to .
Page No 2.34:
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.
Answer:
Given and are the zeros of the quadratic polynomial
We have,
Substituting and then we get,
Hence, the value of is .
Page No 2.34:
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.
Answer:
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
Page No 2.34:
Question 14:
If one zero of the quadratic polynomial f(x) = 4x2 − 8kx − 9 is negative of the other, find the value of k.
Answer:
Since and are the zeros of the quadratic polynomial
= 0
Hence, the Value of is .
Page No 2.34:
Question 15:
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find the quadratic polynomial whose zeros are .
Answer:
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.
Page No 2.34:
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 .
Answer:
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.
Page No 2.34:
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.
Answer:
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.
Page No 2.34:
Question 18:
If α and β are the zeros of the quadratic polynomial f(x) = x2 − p(x + 1) − c, show that (α + 1) (β + 1) = 1 − c.
Answer:
Since and are the zeros of the quadratic polynomial
Then
=
=
We have to prove that
Substituting and we get,
Hence, it is shown that.
Page No 2.34:
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) .
Answer:
(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 .
Page No 2.34:
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.
Answer:
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.
Page No 2.40:
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
Answer:
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.
Page No 2.40:
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.
Answer:
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.
Page No 2.40:
Question 3:
If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Answer:
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 .
Page No 2.40:
Question 4:
Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.
Answer:
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 .
Page No 2.41:
Question 5:
If the zeros of the polynomial f(x) = ax3 + 3bx2 + 3cx + d are in A.P., prove that 2b3 − 3abc + a2d = 0.
Answer:
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 .
Page No 2.41:
Question 6:
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
Answer:
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.
Page No 2.52:
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) = 2x2 − x + 1
(iv) f(x) = 15x3 − 20x2 + 13x − 12, g(x) = 2 − 2x + x2
Answer:
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.
Page No 2.52:
Question 2:
Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm :
(i)
(ii)
(iii)
Answer:
. 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
Page No 2.52:
Question 3:
Obtain all zeros of the polynomial f(x) = 2x4 + x3 − 14x2 − 19x − 6, if two of its zeros are −2 and −1.
Answer:
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.
Page No 2.52:
Question 4:
Obtain all zeros of f(x) = x3 + 13x2 + 32x + 20, if one of its zeros is −2.
Answer:
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 .
Page No 2.52:
Question 5:
Obtain all zeros of the polynomial f(x) = x4 − 3x3 − x2 + 9x − 6, if two of its zeros are
Answer:
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 .
Page No 2.52:
Question 6:
Find all zeros of the polynomial f(x) = 2x4 − 2x3 − 7x2 + 3x + 6, if its two zeros are .
Answer:
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 .
Page No 2.52:
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?
Answer:
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 .
Page No 2.52:
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?
Answer:
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 .
Page No 2.52:
Question 9:
Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120, if two of its zeros are 2 and −2.
Answer:
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 .
Page No 2.52:
Question 10:
Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30, if two of its zeros are .
Answer:
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 .
Page No 2.52:
Question 11:
Find all the zeros of the polynomial 2x3 + x2 − 6x − 3, if two of its zeros are .
Answer:
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.
Page No 2.52:
Question 12:
Find all the zeros of the polynomial x3 + 3x2 − 2x − 6, if two of its zeros are .
Answer:
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 .
Page No 2.53:
Question 1:
Define a polynomial with real coefficients.
Answer:
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
Page No 2.53:
Question 2:
Define degree of a polynomial.
Answer:
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.
Page No 2.53:
Question 3:
Write the standard form of a linear polynomial with real coefficients.
Answer:
Any linear polynomial in variable with real coefficients is of the form, where are real numbers and
Page No 2.53:
Question 4:
Write the standard form of a quadratic polynomial with real coefficients.
Answer:
Any quadratic polynomial in variable with real coefficients is of the form, where are real numbers and
Page No 2.53:
Question 5:
Write the standard form of a cubic polynomial with real coefficients.
Answer:
The most general form of a cubic polynomial with coefficients as real numbers is of the form, where are real number and
Page No 2.53:
Question 6:
Define value of polynomial at a point.
Answer:
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
Page No 2.53:
Question 7:
Define the zero of a polynomial.
Answer:
The zero of a polynomial is defined as any real number such that
Page No 2.53:
Question 8:
The sum and product of the zeros of a quadratic polynomial are and −3 respectively. What is the quadratic polynomial.
Answer:
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
Page No 2.53:
Question 9:
Write the family of quadratic polynomials having and 1 as its zeros.
Answer:
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
Page No 2.53:
Question 10:
If the product of zeros of the quadratic polynomial f(x) = x2 − 4x + k is 3, find the value of k.
Answer:
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.
Page No 2.53:
Question 11:
If the sum of the zeros of the quadratic polynomial f(x) = kx2 − 3x + 5 is 1, write the value of k.
Answer:
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
Page No 2.53:
Question 12:
In Fig. 2.17, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.
Answer:
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
Page No 2.54:
Question 13:
The graph of a polynomial y = f(x), shown in Fig. 2.18. Find the number of real zeros of f(x).
Answer:
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
Page No 2.54:
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.
Answer:
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
Page No 2.54:
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).
Answer:
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
Page No 2.54:
Question 16:
In Q. No. 14, write the sign of c.
Answer:
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
Page No 2.54:
Question 17:
In Q. No. 15, write the sign of c.
Answer:
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
Page No 2.54:
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).
Answer:
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.
Page No 2.54:
Question 19:
If x = 1 is a zero of the polynomial f(x) = x3 − 2x2 + 4x + k, write the value of k.
Answer:
We have to find the value of K if is a zero of the polynomial f(x) = x3 − 2x2 + 4x + k.
Hence, the value of k is
Page No 2.54:
Question 20:
State division algorithm for polynomials.
Answer:
If and are any two polynomials with then we can always find polynomials and such that, where or degree degree
Page No 2.54:
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.
Answer:
Using division algorithm, we have
Hence an example for polynomial,, and satisfying are
Page No 2.54:
Question 22:
Write a quadratic polynomial, sum of whose zeros is and their product is 2.
Answer:
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.
Page No 2.54:
Question 23:
If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.
Answer:
Here represent dividend and represent divisor.
=quadratic polynomial
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
Page No 2.55:
Question 24:
If f(x) = x3 + x2 − ax + b is divisible by x2 − x write the value of a and b.
Answer:
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
Page No 2.55:
Question 25:
If a − b, a and b are zeros of the polynomial f(x) = 2x3− 6x2 + 5x − 7, write the value of a.
Answer:
Let a − b, a and a + b be the zeros of the polynomial then
Sum of the zeros =
Hence, the value of a is.
Page No 2.55:
Question 26:
Write the coefficient of the polynomial p(z) = z5 − 2z2 + 4.
Answer:
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
Page No 2.55:
Question 27:
Write the zeros of the polynomial x2 − x − 6.
Answer:
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
Page No 2.55:
Question 28:
If (x + a) is a factor of 2x2 + 2ax + 5x + 10, find a.
Answer:
Given is a factor of.
Let us now divide by.
We have,
Now, remainder
Hence, the value of a is
Page No 2.55:
Question 29:
For what value of k, −4 is a zero of the polynomial x2 − x − (2k + 2)?
Answer:
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
Page No 2.55:
Question 30:
If 1 is a zero of the polynomial p(x) = ax2 − 3(a − 1) x − 1, then find the value of a.
Answer:
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
Page No 2.55:
Question 31:
If α, β are the zeros of a polynomial such that α + β = −6 and αβ = −4, then write the polynomial.
Answer:
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
Page No 2.55:
Question 32:
If α, β are the zeros of the polynomial 2y2 + 7y + 5, write the value of α + β + αβ.
Answer:
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
Page No 2.55:
Question 33:
For what value of k, is 3 a zero of the polynomial 2x2 + x + k?
Answer:
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
Page No 2.55:
Question 34:
For what value of k, is −3 a zero of the polynomial x2 + 11x + k?
Answer:
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.
Page No 2.55:
Question 35:
For what value of k, is −2 a zero of the polynomial 3x2 + 4x + 2k?
Answer:
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
Page No 2.55:
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)?
Answer:
If a quadratic polynomial is factorized into linear polynomials then the total number of real and distinct zeros of will be.
Page No 2.55:
Question 37:
If a quadratic polynomial f(x) is a square of a linear polynomial, then its two zeros are coincident. (True/False).
Answer:
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.
Page No 2.55:
Question 38:
If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True/False)
Answer:
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. .
Page No 2.55:
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?
Answer:
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
Page No 2.55:
Question 40:
If graph of quadratic polynomial ax2 + bx + c cuts positive direction of y-axis, then what is the sign of c?
Answer:
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
Page No 2.55:
Question 41:
If the graph of quadratic polynomial ax2 + bx + c cuts negative direction of y-axis, then what is the sign of c?
Answer:
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
Page No 2.56:
Question 1:
If α, β are the zeros of the polynomial f(x) = x2 + x + 1, then
(a) 1
(b) −1
(c) 0
(d) None of these
Answer:
Since and are the zeros of the quadratic polynomial
We have
The value of is
Hence, the correct choice is .
Page No 2.56:
Question 2:
If α, β are the zeros of the polynomial p(x) = 4x2 + 3x + 7, then is equal to
(a)
(b)
(c)
(d)
Answer:
Since and are the zeros of the quadratic polynomial
We have
The value of is.
Hence, the correct choice is
Page No 2.56:
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
Answer:
We are given then
One root of the polynomial is reciprocal of the other. Then, we have
1
Hence the correct choice is
Page No 2.56:
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
Answer:
Let, be the zeros of the polynomial and we are given that
=.
Then,
It is given that
=
Substituting , we get
The value of k is.
Hence, the correct alternative is
Page No 2.56:
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) x2 − px + q
(c) qx2 + px + 1
(d) px2 + qx + 1
Answer:
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
Page No 2.56:
Question 6:
If α, β are the zeros of polynomial f(x) = x2 − p (x + 1) − c, then (α + 1) (β + 1) =
(a) c − 1
(b) 1 − c
(c) c
(d) 1 + c
Answer:
Since and are the zeros of quadratic polynomial
We have
The value of is.
Hence, the correct choice is
Page No 2.56:
Question 7:
If α, β are the zeros of the polynomial f(x) = x2 − p(x + 1) − c such that (α +1) (β + 1) = 0, then c =
(a) 1
(b) 0
(c) −1
(d) 2
Answer:
Since and are the zeros of quadratic polynomial
We have
The value of c is.
Hence, the correct alternative is
Page No 2.56:
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
Answer:
If has no real zeros and then
Hence, the correct choice is
Page No 2.56:
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
Answer:
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
Page No 2.57:
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
Answer:
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
Page No 2.57:
Question 11:
If the product of zeros of the polynomial f(x) ax3 − 6x2 + 11x − 6 is 4, then a =
(a)
(b)
(c)
(d)
Answer:
Since and are the zeros of quadratic polynomial
So we have
The value of a is
Hence, the correct alternative is.
Page No 2.57:
Question 12:
If zeros of the polynomial f(x) = x3 − 3px2 + qx − r are in A.P., then
(a) 2p3 = pq − r
(b) 2p3 = pq + r
(c) p3 = pq − r
(d) None of these
Answer:
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
Page No 2.57:
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)
(b)
(c)
(d)
Answer:
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.
Page No 2.57:
Question 14:
If the polynomial f(x) = ax3 + bx − c is divisible by the polynomial g(x) = x2 + bx + c, then ab =
(a) 1
(b)
(c) −1
(d)
Answer:
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 .
Page No 2.57:
Question 15:
If Q.No. 14, c =
(a) b
(b) 2b
(c) 2b2
(d) −2b
Answer:
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
Page No 2.57:
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)
(d) 6
Answer:
If one zero of the polynomial is reciprocal of the other. So
Now we have
Since
Therefore we have
Hence, the correct choice is
Page No 2.57:
Question 17:
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then
(a)
(b)
(c)
(d)
Answer:
We have to find the value of
Given be the zeros of the polynomial
We know that
So
Hence, the correct choice is
Page No 2.57:
Question 18:
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2+ cx + d, then α2 + β2 + γ2 =
(a)
(b)
(c)
(d)
Answer:
We have to find the value of
Given be the zeros of the polynomial
Now
The value of
Hence, the correct choice is
Page No 2.58:
Question 19:
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, then
Answer:
We have to find the value of
Given be the zeros of the polynomial
Now we calculate the expression
Hence, the correct choice is
Page No 2.58:
Question 20:
If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then
(a)
(b)
(c)
(d)
Answer:
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
Page No 2.58:
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)
(b)
(c)
(d)
Answer:
Let and be the zeros of the polynomial
Therefore
The value of
Hence, the correct choice is
Page No 2.58:
Question 22:
If two zeros x3 + x2 − 5x − 5 are , then its third zero is
(a) 1
(b) −1
(c) 2
(d) −2
Answer:
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
Page No 2.58:
Question 23:
The product of the zeros of x3 + 4x2 + x − 6 is
(a) −4
(b) 4
(c) 6
(d) −6
Answer:
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
Page No 2.58:
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
Answer:
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
Page No 2.58:
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
Answer:
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.
Page No 2.58:
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
Answer:
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
Page No 2.58:
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
Answer:
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.
Page No 2.58:
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
Answer:
Let andbe the given zeros and be the third zero of the polynomial . Then,
Substitutingandin
We get
Hence, the correct choice is
Page No 2.58:
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
Answer:
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
Page No 2.58:
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
Answer:
We know that
Therefore,
The polynomial which when divided by gives a quotient and remainder 3, is
Hence, the correct choice is.
View NCERT Solutions for all chapters of Class 10