Question

if f(x)=x⁴-2x³+3x²-9x+b is a polynomial such that when it is divided by x+1 and x+1 , the remainder when f(x) is divided by x-2​

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)=x4−2x3+3x2−5x+8

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

So, Remainder =f(2<