Question

Is (x − 2) a factor of f(x) = x^3 − 2x^2 + 2x + 3? Use either the remainder theorem or the factor theorem to explain your reasoning.

Answer 1

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

Explanation:

By the factor theorem, for any constant
`c`,
`(x - c)` is a factor of polynomial
`f(x)` if and only if
`f(c) = 0`. Note that
`f(c) = 0\!` means that substituting all
`x` in
`f(x)\!` with
`c` and evaluating gives
`0`.

For example, the polynomial in this question is
`f(x) = x^(3) - 2\, x^(2) + 2\, x + 3`. The question is asking whether
`(x - 2)` is a factor of
`f(x)`.

By the factor theorem with
`c = 2`,
`(x - 2)\!` would indeed be a factor of
`f(x)\!` if and only if
`f(2) = 0`. To find the value
`f(2)\!`, simplify replace all "
`x`" in the definition of
`f(x)\!` with
`2`:

`\begin{aligned}f(2) &= 2^(3) - 2* (2^(2)) + 2* 2 + 3 \\ &= 2^(3) - 2^(3) + 7 \\ &= 7\end{aligned}`.

In other words,
`f(2) \\e 0`. By the contrapositive of factor theorem,
`(x - 2)` would not be a factor of
`f(x)`.