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FIND THE ZEROES OF THE QUADRATIC POLYNOMIAL AND VERIFY THE  RELATIONSHIP BETWEEN THE ZEROES AND COFFICIENT OF POLYNOMIAL

 In the general form of quadratic polynomial ax2 + bx + c, there are two zeros say α and β, then;

Sum of the zeros = α + β = -b/a = -(coefficient of x) / (coefficient of x2), and

Product of zeros = α.β = c/a = (Constant Term) / (Coefficient of x2)

Example on Relationship Betrween Zeros and Coefficient of a Polynomial

Example: Find the zeros of the quadratic polynomial x2 + 7x +10, and verify the relationship between the zeroes and the coefficients.

We have,  x2 + 7x +10 = (x +2) (x+5)

So, the value of x2 + 7x + 10 is zero, when x+2 = 0 or x+5 = 0, i.e., when, x = -2 or x = -5. Therefore, the zeroes of x2 + 7x + 10 are -2 and -5.  Now, the relationship between zeros and coefficient of above polynomial can be shown as:-

  Sum of zeroes = -2 + (-5) = -7 = -(7)/1 = -(coefficient of x) / (coefficient of x2)

  Product of zeroes = (-2) x (-5) = 10 =10/1 = (constant Term) / (coefficient of x2

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 find a quadratic polynomial whose zeroes are 1 and-3 and verify the relation between the coefficients and zeroes of the polynomial?

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alfa = 1 

and beta = -3

then alfa+beta = 1-3 =-2

 alfabeta = 1*-3 =-3

since the polynomial is in the form,

x2 - (alfa +beta)x +alfabeta

x2 -(-2)x +(-3)

x2 +2x -3

now , since we know that 

sum of the zeroes are alfa+beta = -b/a = -2

and alfabeta = c/a = -3

HOPE THIS HELPS

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 https://s3mn.mnimgs.com/img/shared/discuss_editlive/1915355/2012_03_26_15_50_41/mathmlequation8797587739562226685.png

The zeros of p (x) are given by p (x) = 0

https://s3mn.mnimgs.com/img/shared/discuss_editlive/1915355/2012_03_26_15_50_41/mathmlequation7987715940584653082.png

https://s3mn.mnimgs.com/img/shared/discuss_editlive/1915355/2012_03_26_15_50_41/mathmlequation6735101308595929437.png

 

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 Relation between the zeroes and coefficient of a polynomial

A polynomial is an algebraic expression consisting of multiple terms. There are various types of polynomials such aslinear, quadratic, cubic……..
A real number k is a zero of a polynomial of p(x) if p(k)= 0.
The general form of linear polynomial is p(x)=ax+b, its zero is –b/a or minus of constant term divided by coefficient of x.

linear, linear polynomial, zero of linear polynomial, relation between the coefficient and zero

General form of quadratic polynomial is ax2 + bx +c.  There are two zeroes of quadratic polynomial.
Factor Theorem: If a is zero of a polynomial p(x) then (x – a) is a factor of p(x).

Sum of zeroes = 2 

Product of zeroes = 3 

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General form of cubic polynomial of ax3 + bx 2+ cx + d where a≠0.  The sum of zeroes of the cubic polynomial =

8 

Sum of the product of zeroes taken two at a time =

9 

Product of zeroes = 4 

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