Factor Theorem Lesson

In the last lesson, we learned about the remainder theorem. Now, we will think about a related concept, known as the factor theorem.

Remainder Theorem

$$f(x)=(x - k)q(x) + r$$ Every polynomial function can be written as some divisor (x - k) times some quotient q(x) plus some remainder r. We saw that if x was equal to k, we get r as the result. $$f(x)=(x - k)q(x) + r$$ $$f(k)=(k - k)q(x) + r$$ $$f(k)=0 + r=r$$ This tells us that evaluating a polynomial function for a value of k will give us r, the remainder from the division of f(x) by (x - k). Additionally, we can say that if r ends up being zero, then (x - k) is a factor or a divisior of f(x). Let's think about this with an example.
Example #1: Determine if g(x) is a factor of f(x). $$f(x)=x^4 + 5x^3 - 45x^2 + 63x$$ $$g(x)=x - 3$$ We just need to check if f(3) = 0. $$f(3)=(3)^4 + 5(3)^3 - 45(3)^2 + 63(3)$$ $$f(3)=81 + 135 - 405 + 189$$ $$f(3)=0$$ Since f(3) is 0, we know that g(x) is a factor of f(x).

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