Exercise 2.4 Class 10 maths chapter 2. Polynomials
Q1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) 2x3 + x2 - 5x + 2; α = ½, β = 1 and γ = – 2;
(ii) x3 – 4x2 + 5x – 2; α = 2, β = 1, and γ = 1
Solution: (i) 2x3 + x2 - 5x + 2; α = ½, β = 1 and γ = – 2;
Given: p(x) = 2x3 + x2 - 5x + 2
p(1) = 2(1)3 + (1)2 - 5(1) + 2
= 2 + 1 - 5 + 2
= 5 - 5 = 0
p(x) = 0
Therefore, 1 is the zero of p(x)
अब, p(-2) = 2(-2)3 + (-2)2 - 5(-2) + 2
= -16 + 4 + 10 + 2
= 16 - 16 = 0
p(x) = 0
Therefore, -2 is the zero of p(x)
So, α = ½, β = 1 and γ = – 2 are zeroes.
And coefficients a = 2, b = 1, c = - 5 and d = 2
Verification of relation between zeroes and coefficients:
Verified by equation (1) (2) and (3)
Solution: (ii) x3 – 4x2 + 5x – 2; α = 2, β = 1, and γ = 1
Given that: p(x) = x3 – 4x2 + 5x – 2
p(2) = (2)3 – 4(2)2 + 5(2) – 2
= 8 – 4.4 + 10 – 2
= 8 – 16 + 10 – 2
= 18 – 18 = 0
Hence, α = 2 is the zero of p(x)
Now for β = 1
p(x) = x3 – 4x2 + 5x – 2
p(1) = (1)3 – 4(1)2 + 5(1) – 2
= 1 – 4.1 + 5 – 2
= 1 – 4 + 5 – 2
= 6 – 6 = 0
Hence, β = 1 is the zero of p(x)
Now, for γ = 1
p(x) = x3 – 4x2 + 5x – 2
p(1) = (1)3 – 4(1)2 + 5(1) – 2
= 1 – 4.1 + 5 – 2
= 1 – 4 + 5 – 2
= 6 – 6 = 0
Hence, γ = 1 is the zero of p(x)
So, α = 2, β = 1 and γ = 1 are zeroes.
And coefficients a = 1, b = – 4, c = 5 and d = – 2
Verification of relation between zeroes and coefficients: