Question

What is the remainder R when the polynomial p(x) is divided by (x + 1)? Is (x + 1) a factor of p(x)? p(x) = -3x4 + 2x3 - x2 + 6

Answer 1

(x+1) is a factor with remainder 0.

Explanation:

We divide (x+1) into the polynomial
`-3x^4+2x^3-x^2+6` through long division or synthetic. We choose long division and look for what will multiply with (x+1) to make the polynomial
`-3x^4+2x^3-x^2+6` .

`(x+1)(-3x^3)=-3x^4-3x^3`

We subtract this from the original
`-3x^4-(-3x^4)+2x^3-(-3x^3)-x^2+6`.

This leaves
`5x^3-x^2+6`. We repeat the step above.

`(x+1)(5x^2)=5x^3+5x^2`.

We subtract this from
`5x^3-(5x^3)-x^2-(5x^2)+6=-6x^2+6`. We repeat the step above.

`(x+1)(-6x)=-6x^2-6x`.

We subtract this from
`-6x^2-(-6x^2)+0x-(-6x)+6=6x+6`. We repeat the step above.

`(x+1)(6)=-6x+1`.

We subtract this from
`6x-(6x)+6-(6)=0`. There is no remainder. This means (x+1) is a factor.