Question

f(x)=x4-2x3-ax+b is a polynomial such that when it is divided byx-1 and x+1 the remainder are 5,19 respectively.then the remainder when f(x) is divided by (x-2) is

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Answer 1

• mate here we used polynomial remainder theorem.

1. a polynomial f(x) is divided by a linear polynomial and leaves reminder which equal f(a)

• and the question is .$=>x^{4}-2x^{3}-ax+b.$

and ,

• when it is divided by (x-2).
• it's remainder (f1)= 5.

$f(1) =1^{4}+2*1^{3}+3*1^{3}-a*{1}+b =5 \\ => 1-2+3+a+b =5 \\ =>a-b= -3 ....... equation (1)$

【 (*) this sign show there is a multiply symbol】

now like that,

$f(1)=> (-1)^{4}+2*(-1)^{3}×3*(-1)^{2}-a*(-1)+b =19 \\ f(1) => 1+2+3+a+b =19 \\ f1 =>a+b =13 ..... equation(2)$

• now adding equation 1 and 2..

• we get,

➡️ (a+b)+(a-b)=(-3)+13

➡️ a+b+a-b = 10. (b-b cancel out with each other)

➡️ 2a =10

. ➡️ a =10/2

➡️a = 5

so,

➡️a+b=13

➡️5+b=13

➡️b= 13-5

➡️b=8

therefore :- a = 5 , b = 8 ANSWER

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