If fx=x4-2x^3+3x^2-ax+b is divided by x-1 and x+1,it leaves remainder 5 and 19 respectively.Find the value of a and b.
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Answers
Given that the equation
(x) = x4 – 2x3 + 3x2 – ax +b
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 ——- (1)
According to given condition f(1) = 5
f(x) = x4 – 2x3 + 3x2 – ax
⇒ 14 – 2 3 + 3 2 – a (1) b = 5
⇒ 1 – 2 + 3 – a + b = 5
∴ b – a = 3 —— (2)
solving equations (1) and (2)
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
Also f(x) is divided by (x-3) so remainder will be f(3)
∴ f(x)= x4 – 2x3 + 3x2 – 5x + 8
⇒ f(3) = 34 – 2 × 33 + 3 × 32 – 5 × 3 + 8
= 81 – 54 + 27 – 15 + 8
= 47
Therefore, f(x) = x4 – 2x3 + 3x2 – ax +b when a=3 and b= 8 is 47.
Hope it helps :)
Answer:
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