Question

The polynomial p(x) =x⁴-2x³+3x²-ax+b when divided by (x-1) and (x+1) leaves the remainder 5 and 19 respectively . Find the values of a and b .Hence find the remainder when p(x) is divided by (x-2). Please give me the answer fast .​

Answer 1

Solution :-

we know that, if p(x) is divide by (x - a) it will gives remainder as p(a) .

so,

→ p(x) = x⁴-2x³+3x²-ax+b

when divide by (x - 1)

→ p(1) = 5

→ (1)⁴ - 2(1)³ + 3(1)² - a*1 + b = 5

→ 1 - 2 + 3 - a + b = 5

→ 2 - a + b = 5

→ b - a = 3 ------- Eqn.(1)

similarly,

when divide by (x + 1)

→ p(-1) = 19

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

→ 1 + 2 + 3 + a + b = 19

→ 6 + a + b = 19

→ a + b = 13 ------- Eqn.(2)

adding Eqn.(1) and Eqn.(2),

→ (b - a) + (a + b) = 3 + 13

→ 2b = 16

→ b = 8 .

then,

→ 8 - a = 3

→ a = 5.

therefore,

→ p(x) = x⁴ - 2x³ + 3x² - 5x + 8

hence, when divide by (x - 2)

→ p(2) = (2)⁴ - 2(2)³ + 3(2)² - 5*2 + 8

→ p(2) = 16 - 16 + 12 - 10 + 8

→ p(2) = 10 (Ans.)

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