Question

What must be added to the polynomial f(x) =x⁴ + 2x³ - 2x² + x - 1 so that the resulting polynomial is exactly divisible by x² + 2x - 3 ? ​

Answer 1

Your answer is -;:::::

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..

Answer 2

Given:

f(x) =x⁴ + 2x³ - 2x² + x - 1

To find:

What must be added to the polynomial so that the resulting polynomial is exactly divisible by x² + 2x - 3

STEP BY STEP EXPLANATION:

➭ the given polynomial is

${x}^{4} + {2x}^{3} - {2x}^{2} + x - 1$

➭ Divisor = ${x}^{2} + 2x - 3$

➭ Remainder = - x + 2

➭ therefore,we should add - ( - x + 2 ) to make it divisible exactly by ${x}^{2} + 2x - 3$

∴ Thus,we should add x - 2 to

${x}^{4} +{2x}^{3} - {2x}^{2} + x - 1$

Hence verified !