Question

1. The polynomial p(x)=x* - 2x +3r? - ax +b when divided by (x - 1) and
(x + 1) leaves the remainders 5 and 19 respectively. Find the values of a
and b. Hence, find the remainder when p(x) is divided by (x-2).​

Answer 1

Correct Question:-

When f(x) = x4 - 2x3 + 3x2 - ax is divided by x + 1 and x - 1 , we get remainders as 19 and 5 respectively .

Find the remainder if f(x) is divided by x - 3 .

Solution:-

When f(x) is divided by (x+1) and (x-1) , the remainders are 19 and 5 respectively .

∴ f(-1) = 19 and f(1) = 5

⇒ (-1)4 - 2 (-1)3 + 3(-1)2 - a (-1) + b = 19

⇒ 1 +2 + 3 + a + b = 19

∴ a + b = 13 ------- (i)

Again , f(1) = 5

⇒ 14 - 2 × 13 + 3 × 12 - a × 1 b = 5

⇒ 1 - 2 + 3 - a + b = 5

∴ b - a = 3 ------ (ii)

solving eqn (i) and (ii) , we get

a = 5 and b = 8

Now substituting the values of a and b in f(x) , we get

∴ f(x) = x4 - 2x3 + 3x2 - 5x + 8

Now f(x) is divided by (x-3) so remainder will be f(3)

∴ f(x) = ∴ f(x) = x4 - 2x3 + 3x2 - 5x + 8

⇒ f(3) = 34 - 2 × 33 + 3 × 32 - 5 × 3 + 8

= 81 - 54 + 27 - 15 + 8 = 47

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