Question

What must be added to the polynomial $f(x)=x^{4}+ 2x^{3}-2x^{2}+x-1$ so that the resulting polynomial is exactly divisible by $x^{2}+ 2x-3?$

Answer 1

A polynomial is exactly divisible by another polynomial, if remainder is zero. So , here find the remainder and add negative of remainder in f(x) , so that the resulting polynomial is divisible by g(x).

Given : f(x) = x⁴ + 2x³ -2x² + x -1

and g(x) = x² + 2x + 3

Now in Dividing f(x) by g(x) , we get the following division process.  

DIVISION PROCESS  is in the attachment.

Here, Remainder is -x+2 . Now the polynomial  f(x) = x⁴ + 2x³ -2x² + x -1 will be exactly divisible by g(x) = x² + 2x + 3, when reminder is zero.  So to make the remainder 0 , x+2 is to be added in f(x) .  

Hence, if we add x + 2 in f(x) , then it will be divisible by g(x) = x² + 2x + 3.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

Answer:

ANSWER

We know, p(x)=[g(x)×q(x)]+r(x)

∴p(x)−r(x)=g(x)×q(x)

∴p(x)+{−r(x)}=g(x)×q(x)

It is clear that RHS is divisible by g(x). ∴LHS is also divisible by g(x)

Thus, if we add −r(x) to p(x), then the resulting polynomial is divisible by g(x).

Let us divide p(x)=x

4

+2x

3

−2x

2

+x−1 by g(x)=x

2

+2x−3 to find the remainder r(x).

∴r(x)=−x+2⇒{−r(x)}=x−2

Hence, we should add (x−2) to p(x)=x

4

+2x

3

−2x

2

+x−1 so that the resulting polynomial is exactly divisible by x

2

+2x−3.