Factor Theorem
The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors.
the factor theorem
According to the factor theorem, [latex]k[/latex] is a zero of [latex]f(x)[/latex] if and only if [latex](x−k)[/latex] is a factor of [latex]f(x)[/latex].
Let’s walk through the proof of the theorem.
[latex]f(x) = (x - k)q(x) + r[/latex]
If [latex]k[/latex] is a zero, then the remainder [latex]r[/latex] is [latex]f(k) = 0[/latex] and [latex]f(x) = (x - k)q(x) + 0[/latex] or [latex]f(x) = (x - k)q(x)[/latex].
Notice, written in this form, [latex]x - k[/latex] is a factor of [latex]f(x)[/latex]. We can conclude if [latex]k[/latex] is a zero of [latex]f(x)[/latex], then [latex]x - k[/latex] is a factor of [latex]f(x)[/latex].
Similarly, if [latex]x - k[/latex] is a factor of [latex]f(x)[/latex], then the remainder of the Division Algorithm [latex]f(x) = (x - k)q(x) + r[/latex] is [latex]0[/latex]. This tells us that [latex]k[/latex] is a zero.
This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree [latex]n[/latex] in the complex number system will have [latex]n[/latex] zeros. We can use the Factor Theorem to completely factor a polynomial into the product of [latex]n[/latex] factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.
- Use synthetic division to divide the polynomial by [latex](x-k)[/latex]
- Confirm that the remainder is [latex]0[/latex].
- Write the polynomial as the product of [latex](x-k)[/latex] and the quadratic quotient.
- If possible, factor the quadratic.
- Write the polynomial as the product of factors.
