Question

What must be added to polynomial f(x) = x4 + 2x2 – 2x2 + x – 1 so that resulting
polynomial is exactly divisible by x2 + 2x – 3?

Answer 1

Step-by-step explanation:

Let P(x)=X^4+2X^3–2X^2+X-1

G(x)=X^2+2X-3

=>X^2+(3–1)X-3

=>X^2+3X-X-3

=>X(X+3)-1(X+3)

=>(X+3)(X-1)

If ax+b added to P(x), then P(x) must be divisible by G(x)

Therefore(x+3) and(x-1) will be the factor of

X^4+2X^3–2X^2+X-1+aX+b

=>(-3)^4+2(-3)^3–2(-3)^2+(-3)-1+a(-3)+b

=>81–54–18–3–1–3a+b=0.

=>5–3a+b=0…….…..…(1)

(1)^4+2(1)^3–2(1)^2+1–1+a+b=0

1+2–2+1–1+a+b=0

1+a+b=0……………(2)

Subtracting(1) from(2)

-4+4a=0

=>a=1

b=-2

So x-2 must be added