Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

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.

Sagot :

Answer:

[tex](x - 2)[/tex] isn't a factor of [tex]f(x) = x^{3} - 2\, x^{2} + 2\, x + 3[/tex].

Step-by-step explanation:

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

For example, the polynomial in this question is [tex]f(x) = x^{3} - 2\, x^{2} + 2\, x + 3[/tex]. The question is asking whether [tex](x - 2)[/tex] is a factor of [tex]f(x)[/tex].

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

[tex]\begin{aligned}f(2) &= 2^{3} - 2\times (2^{2}) + 2\times 2 + 3 \\ &= 2^{3} - 2^{3} + 7 \\ &= 7\end{aligned}[/tex].

In other words, [tex]f(2) \ne 0[/tex]. By the contrapositive of factor theorem, [tex](x - 2)[/tex] would not be a factor of [tex]f(x)[/tex].