Question

How to solve this polynomial equation?

When S(x)= `x^4+ax^3+bx^2+4x-7` is divided by `(x+1)` `(x-1)`, the remaider is `-2x+5`. Find a and b.

Answer 1

I get `a=-6, b=-13`.

Explanation:

Since we know the remainder is `-2x+5`, we are sure that if we subtract that from the given polynomial then the difference will be exactly a multiple of `(x+1)(x-1)=x^2-1`. So we render this:

`(x^4+ax^3+bx^2+4x-7)-(-2x+5)=x^4+ax^3+bx^2+6x-12=(x^2-1)(x^2+px+q)`

Then we multiply out that last product and match like power terms:

`(x^2-1)(x^2+px+q)=x^4+px^3+(q-1)x^2-px-q=x^4+ax^3+bx^2+6x-12`

Then match like powers:

`x^4=x^4`

`px^3=ax^3`

`bx^2=(q-1)x^2`

`6x=-px`

`-12=q`

So `p=-6` meaning `a=p=-6`, and `q=-12` meaning `b=q-1=-13`.