AnswWhen 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=10er:
Step-by-step explanation:
