Question

what must be added to the polynomial f(x)=x^4+2x^3-2x^2+x-1 to make it divisible by x² + x - 2​

Answer 1

Step-by-step explanation:

Solution :-

Given polynomial is f(x) = x⁴+2x³-2x²+x-1

Given divisor = x²+x-2

On dividing f(x) by the given divisor then

x²+x-2 ) x⁴+2x³-2x²+x-1 ( x²+x-1

x⁴+x³-2x²

(-)

_____________

x³+0 +x

x³+x²-2x

(-)

______________

-x²+3x -1

-x²- x +2

(-)

______________

4x-3

______________

We get the quotient = x²+x-1

Remainder = 4x-3

We know that

If the polynomial is divisible by the another polynomial then the remainder is zero.

So we should add -(4x-3) = -4x+3 to f(x) .

Answer :-

-4x+3 should be added to f(x) to make it divisible by x²+x-2.

Check :-

If we add -4x+3 then f(x) becomes

f(x) = x⁴+2x³-2x²+x-1-4x+3

=> f(x) = x⁴+2x³-2x²-3x+2

g(x) = x²+x-2.

x²+x-2) x⁴+2x³-2x²-3x+2 ( x²+x-1

x⁴+x³-2x²

(-)

_____________

x³+0 -3x

x³+x²-2x

(-)

______________

-x²-x +2

-x²- x +2

(-)

______________

0

______________

f(x) is divisible by x²+x-2

Verified the given relations in the given problem.