Question

if p(x)=x^4-2x^3+3x^2-ax+8 is a polynomial such that when it is divided by (x-1) , the remainder is 5. determine the remainder when p(x) is divided by (x-2) and (x+1)​

Answer 1

Answer:

when f(x) is divided by x-1 and x+1 the remainder are 5 and 19 respectively.

∴f(1)=5 and f(−1)=19

⇒(1)

4

−2×(1)

3

+3×(1)

2

−a×1+b=5

and (−1)

4

−2×(−1)

3

+3×(−1)

2

−a×(−1)+b=19

⇒1−2+3−a+b=5

and 1+2+3+a+b=19

⇒2−a+b=5 and 6+a+b=19

⇒−a+b=3 and a+b=13

adding these two equations, we get

(−a+b)+(a+b)=3+13

⇒2b=16⇒b=8

putting b=8 and −a+b=3, we get

−a+8=3⇒a=−5⇒a=5

putting the values of a and b in

f(x)=x

4

−2x

3

+3x

2

−5x+8

the remainder when f(x) is divided by (x-2) is equal to f(2).

so, remainder =f(2)=(2)

4

−2×(2)

3

+3×(2)

2

−5×2+8=16−16+12−10+8=10