Explain the working of this question
What does it mean to explain the relationship between zeroes and coefficients
class 10th polynomials 2nd exercise rd Sharma
mehakNewbie
1. Verify that the numbers given alongside the cubic polynomials below are their zeroes. Also, verify the relationship between the zeros and coefficients in each of the following cases: (i) f(x) = 2×3 + x2 – 5x + 2; 1/2, 1, -2
Share
Solution:
Given, f(x) = 2x3 + x2 – 5x + 2, where a= 2, b= 1, c= -5 and d= 2
For x = 1/2
f(1/2) = 2(1/2)3 + (1/2)2 – 5(1/2) + 2
= 1/4 + 1/4 – 5/2 + 2 = 0
⇒ f(1/2) = 0, hence, x = 1/2 is a root of the given polynomial.
For x = 1
f(1) = 2(1)3 + (1)2 – 5(1) + 2
= 2 + 1 – 5 + 2 = 0
⇒ f(1) = 0, hence, x = 1 is also a root of the given polynomial.
For x = -2
f(-2) = 2(-2)3 + (-2)2 – 5(-2) + 2
= -16 + 4 + 10 + 2 = 0
⇒ f(-2) = 0, hence, x = -2 is also a root of the given polynomial.
Now,
Sum of zeros = -b/a
1/2 + 1 – 2 = – (1)/2
-1/2 = -1/2
Sum of the products of the zeros taken two at a time = c/a
(1/2 x 1) + (1 x -2) + (1/2 x -2) = -5/ 2
1/2 – 2 + (-1) = -5/2
-5/2 = -5/2
Product of zeros = – d/a
1/2 x 1 x (– 2) = -(2)/2
-1 = -1
Hence, the relationship between the zeros and coefficients is verified.