Question

When f(x) = x4 – 2x3 + 3x2 - ax + b is divided by x+1 and x – 1, we get reminders as 19 and 5 respectively, find the values of a and b. Hence find the remainder if f(x) is divided by x – 2.

Answer 1

Step-by-step explanation:

p(x)=x^4–2x^3+3x^2-ax+b

By remainder theorem, when p(x) is divided by (x-1) and (x+1) , the remainders are equal to p(1) and p(-1) respectively.

By the given condition, we have

p(1)=5 and p(-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

=> -a+b=5–1+2–3 and 1+2+3+a+b=19

=> -a+b=3 and a+b=19–1–2–3

=> -a+b=3 and a+b=13

Adding these two equations,we get

-a+b+a+b=3+13

=> 2b=16

=> 2b/2=16/2

=> b=8

Putting b=8 in a+b=13 , we get

a+8=13

=> a=13–8

=> a=5

Therefore, a=5 and b=8 .

Answer 2

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=5and (−1) 4 −2×(−1) 3 +3×(−1) 2 −a×(−1)+b=19⇒1−2+3−a+b=5and 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=8and−a+b=3,weget−a+8=3⇒a=−5⇒a=5

Putting the values of a and b inf(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