Question

The polynomial f(x)= x4 - 2 * x3+a*x + b when divided by (x-1) and (x+1) leaves the remainider 5 and 19 respectively. Find the values of a and b hence find the remainder when f(x) is divided be (x-3)

Answer 1

f(x)= x⁴-2x³+ax+b= 5

g(x)=x-3
x=3

p(x)= x-1
x=1

q(x)= x+1
x= -1

f(1)= 1⁴-2(1)³+a(1)+b=5
= 1-2+a+b=5
= -1+a+b=5
=a+b=5+1
=a+b=6 ———— 1

f(-1)= (-1)⁴-2(-1)³+a(-1)+b=19
=1+2-a+b=19
=3-a+b=19
=b-a=19-3
=b-a=16 ———— 2

Subtracting 1 from 2
(b-a)-(a+b)=16-6
b-a-a-b= 10
-2a= 10
a= 10÷-2
a=-10÷2
a=-5

Using 1
b-a=16
b-(-5)= 16
b+5=16
b=16-5
b=11

a = -5
b = 11

f(x)=x⁴-2x³+ax+b
f(3)=3⁴-2(3)³+(-5)3+11
=81-54-15+11
=23

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