Question

Determine whether the given binomial is a factor of the polynomial p(x) . p(x)=x^ 3 +2x^ 2 -x-2;(x+2)

Answer 1

`(x + 2)` isn't a factor of the polynomial
`p(x) = x^(3) + 2\, x^(2) - x - 2`.

Explanation:

By the Factor Theorem, for any constant
`a`,
`(x - a)` would be a factor of a polynomial
`p(x)` if and only if
`x = a` is a root (a zero) of that polynomial (
`p(a) = 0`.)

In other words,
`(x - a)` is a factor of polynomial
`p(x)` if and only if replacing all mentions of
`x` in
`p(x)\!` with the constant
`a` would yield
`0`.

For example, in this question,
`(x + 2)` could be rewritten as
`(x - (-2))` (value of the constant is
`a = (-2)`.)

By the Factor Theorem,
`(x - (-2))` would be a factor of the polynomial
`p(x) = x^(3) + 2\, x^(2) - x - 2` if and only if replacing all mentions of
`x` in
`x^(3) + 2\, x^(2) - x - 2` with
`(-2)` evaluates to
`0`.

Replacing all mentions of
`x` in
`x^(3) + 2\, x^(2) - x - 2` with
`(-2)` gives:

`(-2)^(3) + 2\, (-2)^(2) - (-2) - 2`.

Simplify:

`\begin{aligned}& (-2)^(3) + 2\, (-2)^(2) - (-2) - 2 \\ =\; & (-8) + 8 + 2 - 2 \\ =\; & 0\end{aligned}`.

Since
`p(-2) = 0`,
`(x - (-2))` (or equivalently,
`(x + 2)`) is indeed a factor of the polynomial
`p(x)`.