When f(x)=x^4-2x^3+3x^2-ax+b is divided by x+1 and x-1 we get remainders 19 and 5 respectively .Find the remainder when f(x) is divided by x-3.

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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) = x^{4 -}2x³+ 3x² - 5x + 8

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

∴ f(x) = ∴ f(x) = x^{4 -}2x³+ 3x² - 5x + 8

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

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

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