Question

Is x + 1 a factor of
`x^(4) +2x^(3) +2x^(2) -2x-3`?

Answer 1

Yes x+1 is a factor of that

Answer 2

Yes.

Explanation:

We want to determine if (x + 1) is a factor of the polynomial:

`x^4 + 2x^3 + 2x^2 - 2x -3`

According to the Factor Theorem, if a binomial in the form (x - a) is a factor of a polynomial P(x), then P(a) must equal zero.

Our binomial factor is (x + 1) or (x - (-1)). Hence, a = -1.

Let our polynomial be P(x). Find P(-1):

`\displaystyle\begin{aligned} P(-1) &= (-1)^4 + 2(-1)^3 +2(-1)^2 -2(-1) -3 \\&=0\end{aligned}`

Therefore, since the resulting value is indeed zero, (x + 1) is indeed a factor of the given polynomial.

In conclusion: yes.