Relation between the zero and the coefficients of a polynomials Mathematics, knowledgeuniverseonline.com

Chapter : Polynomials

Relation between the zero and the coefficients of a polynomials

Consider quadratic polynomial P(x) = 2x² – 16x + 30.
Now, 2x² – 16x + 30
= (2x – 6) (x – 3)
= 2(x – 3) (x – 5)
The zeroes of P(x) are 3 and 5.

Sum of the zeroes are = 3 + 5 = 8 ==

Product of the zeroes are = 3 × 5 = 15 = =

So if ax² + bx + c, a ≠ 0 is a quadratic polynomial and a, b are two zeroes of polynomial then

Example: Find the zeroes of the quadratic polynomial 6x² – 13x + 6 and verify the relation between the zeroes and its coefficients.

Solution: We have,
6x² – 13x + 6
= 6x² – 4x – 9x + 6
= 2x (3x – 2) –3 (3x – 2)
= (3x – 2) (2x – 3)
So, the value of quadratic equation 6x² – 13x + 6 is 0, when
(3x – 2) = 0 or (2x – 3) = 0 i.e.,
When x = or
Therefore, the zeroes of 6x² – 13x + 6 are and .

Sum of the zeroes :

+ = = =

Product of the zeroes
= × = =

Example: Find the zeroes of the quadratic polynomial 4x2 – 9 and verify the relation between the zeroes and its coefficients.

Solution: We have,
4x² – 9
= (2x)² – 3²
= (2x – 3) (2x + 3)
So, the value ofa quadratic equation 4x² – 9 is 0, when
2x – 3 = 0 or 2x + 3 = 0
i.e., when x = or x = .
Therefore, the zeroes of 4x² – 9 are & .