KSEEB Solutions for Class 10 Maths Chapter 9 Polynomials Ex 9.4

KSEEB Solutions for Class 10 Maths Chapter 9 Polynomials Ex 9.4

Karnataka State Syllabus Class 10 Maths Chapter 9 Polynomials Ex 9.4

Question 1.
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; 1/2, 1, – 2
(ii) x³ – 4x² + 5x – 2; 2, 1, 1
Answer:
Let P(x) = 2x³ + x² – 5x + 2 on comparing with general polynomial P(x) = ax³ + bx² + cx + d we get
a = 2, b = 1, c = – 5 and d = 2
Given zeroes 1/2, 1, – 2

P(1) = 2 (1)³ + (1)² – 5(1) + 2
= 2 × 1 + 1 – 5 + 2
= 2 + 1 – 5 + 2 = 5 – 5 = 0
P(1) = 0
P(- 2) = 2(- 2)³ + (-2)² – 5(- 2) + 2
= 2 × – 8 + 4 + 10 + 2
= – 16 + 16
P(- 2) = 0
Hence 1/2, 1 & -2 are the zeroes of the given cubic polynomial
Again consider α = 1/2, β = 1 & γ = – 2

But remainder is given to be x + a comparing coefficients of x in the remainder obtained and remainder given:
(2K – 9)x – K(8 – K)+ 10 = x + a
∴ 2K – 9 = 1
∴ 2K = 1 + 9
K = 10/2
K = 5 and – K(8 – K) + 10 = a
Put K = 5
– 5(8 – 5) + 10 = a
– 5(3) + 10 = a
– 15 + 10 = a
a = – 5