If f(x) = x4 2x3 + 3x2 ax + b is divided by (x 1) and (x + 1), it leaves the remainders 5 and 19 respectively. Find a and
b.
Answers
f(x) = x-1 =0
x= 1
put x = 1
1+2(1)+3(1)+a(1)+b =0
1+2+3+a+b=0
6+a+b=0
a+b= -6
a=-6-b
first equation
let
x+1 =0
x= -1
put x=-1
1+2(-1)+3(1)+a(-1)+b=0
1-2+3-a+b =0
-4-a+b =0
-a+b = 4
b=4+a
second equation
from 1 and 2 eq.
a=-6-b
a=-6-(4+a)
a= -6-4-a
a=-10-a
a+a =-10
2a =-10
a=-5
putting value of a in 2 eq.
b=4+a
b= 4+(-5)
b= -1
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