Question

Is (x − 2) a factor of f(x) = x3 − 2x2 + 2x + 3? Use either the remainder theorem or the factor theorem to explain your reasoning.

Answer 1

(x - 2) is not a factor of f(x)

Explanation:

We can use the Remainder Theorem to check if (x - 2) is a factor of f(x) = x^3 - 2x^2 + 2x + 3.

The Remainder Theorem states that if we evaluate a polynomial at a specific value and the result is zero, then that value must be a root of the polynomial and therefore the polynomial can be divided by the corresponding linear factor.

So, if we plug in x = 2 into f(x), we get:

f(2) = 2^3 - 2 * 2^2 + 2 * 2 + 3

= 8 - 8 + 4 + 3

= 7

Since f(2) is not equal to zero, it follows that (x - 2) is not a factor of f(x).

Alternatively, we could use the Factor Theorem, which states that a polynomial is divisible by (x - a) if and only if f(a) = 0. Since f(2) is not equal to zero, (x - 2) is not a factor of f(x).