Question

If f(x) = x³ + px + q is divisible by x² + x - 2 then the remainder when f(x) is divided by x + 1 is:
(A) 4
(B) 3
(C)-4
(D) 1​

Answer 1

GIVEN :

The polynomial $f(x) = x^3 + px + q$ is divisible by $x^2 + x - 2$

TO FIND :

The remainder when the polynomial f(x) is divided by x + 1

SOLUTION :

Given that the polynomial $f(x) = x^3 + px + q$ is divisible by $x^2 + x - 2$

It can be written as  below:

$x^3 + px + q=(x^2+x-2)(x+a)$

By using the Distributive property:

a(x+y+z)=ax+ay+az

$x^3+px+q=x^2(x)+x^2(a)+x(x)+x(a)-2(x)-2(a)$

$x^3+0x^2+px + q=x^3+ax^2+x^2+ax-2x-2a$

$x^3+0x^2+px + q=x^3+(a+1)x^2+(a-2)x-2a$

Now equating the coefficients of $x^2$ on both the sides we get

0=a+1

a+1=0

∴ a=-1

Substitute the value of a=-1 in $x^3+px+q=(x^2+x-2)(x+a)$

∴ $x^3+px+q=(x^2+x-2)(x-1)$

$x^3+px+q=(x^2+x-2)(x-1)$

Substitute the value of a=-1 in $(x^2+x-2)(x-1)$,

$=((-1)^2+(-1)-2)(-1-1)$

$=(1-1-2)(-2)$

=(-2)(-2)

=4

∴ The remainder when the polynomial f(x) is divided by x+1 is 4.

Option A) 4 is correct.

Answer 2

Answer:

As the other guy said,

Option A) 4 is correct

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