Lesson Objectives

- Demonstrate an understanding of synthetic division
- Demonstrate an understanding of the remainder theorem
- Learn how to use the factor theorem to determine if (x - k) is a factor of f(x)

## How to Determine if (x - k) is a Factor Using the Factor Theorem

In the last lesson, we did a brief review of synthetic division, and we learned about the remainder theorem. In this lesson, we will learn about the factor theorem, which is a direct result of the remainder theorem.

The polynomial (x - k) is a factor of the polynomial f(x) if and only if f(k) = 0. In other words, k is a zero of f(x) if and only if (x - k) is a factor of f(x). Let's look at some examples.

Example #1: Determine if g(x) is a factor of f(x). $$f(x)=x^4 + 5x^3 - 45x^2 + 63x$$ $$g(x)=x - 3$$ $$k = 3$$We just need to check if f(3) = 0. We can do this by plugging in or with synthetic division.

Plug in Method: $$f(3)=(3)^4 + 5(3)^3 - 45(3)^2 + 63(3)$$ $$=81 + 135 - 405 + 189 = 0$$ Synthetic Division Method: Since f(3) = 0, we know that g(x) is a factor of f(x).

Example #2: Determine if g(x) is a factor of f(x). $$f(x) = x^5 + 2x^4 - 9x^2 - 3x$$ $$g(x) = x + 1$$ We want g(x) to be of the form x - k: $$g(x) = x - (-1)$$ $$k = -1$$ We just need to check if f(-1) = 0. We can do this by plugging in or with synthetic division.

Plug in Method: $$f(-1)=(-1)^5 + 2(-1)^4 - 9(-1)^2 - 3(-1)$$ $$=-1 + 2 - 9 + 3 = -5$$ Synthetic Division Method: Since f(-1) = -5, we know that g(x) is not a factor of f(x).

Example #3: Factor each completely. $$f(x) = x^3 + 3x^2 - 49x + 45$$ $$f(-9) = 0$$ Here we are given the fact that f(-9) = 0, which means that [x - (-9)] or (x + 9) is a factor of f(x). Let's set up a synthetic division. $$\frac{x^3 + 3x^2 - 49x + 45}{x + 9} = x^2 - 6x + 5$$ Now, we just need to factor x

### Remainder Theorem

For any polynomial f(x) and any complex number k, there exists a unique polynomial q(x) and a number r such that: $$f(x)=(x - k)q(x) + r$$ Replacing x with k: $$f(k) = (k - k)q(k) + r = 0 + r = r$$ The result here, known as the remainder theorem, tells us that f(k) = r. In other words, if the polynomial f(x) is divided by (x - k), then the remainder is equal to f(k).### Factor Theorem

Additionally, we learned that if f(k) = 0, then the remainder when f(x) is divided by (x - k) is 0. This tells us that (x - k) is a factor of f(x). This result is known as the factor theorem, which is given as:The polynomial (x - k) is a factor of the polynomial f(x) if and only if f(k) = 0. In other words, k is a zero of f(x) if and only if (x - k) is a factor of f(x). Let's look at some examples.

Example #1: Determine if g(x) is a factor of f(x). $$f(x)=x^4 + 5x^3 - 45x^2 + 63x$$ $$g(x)=x - 3$$ $$k = 3$$We just need to check if f(3) = 0. We can do this by plugging in or with synthetic division.

Plug in Method: $$f(3)=(3)^4 + 5(3)^3 - 45(3)^2 + 63(3)$$ $$=81 + 135 - 405 + 189 = 0$$ Synthetic Division Method: Since f(3) = 0, we know that g(x) is a factor of f(x).

Example #2: Determine if g(x) is a factor of f(x). $$f(x) = x^5 + 2x^4 - 9x^2 - 3x$$ $$g(x) = x + 1$$ We want g(x) to be of the form x - k: $$g(x) = x - (-1)$$ $$k = -1$$ We just need to check if f(-1) = 0. We can do this by plugging in or with synthetic division.

Plug in Method: $$f(-1)=(-1)^5 + 2(-1)^4 - 9(-1)^2 - 3(-1)$$ $$=-1 + 2 - 9 + 3 = -5$$ Synthetic Division Method: Since f(-1) = -5, we know that g(x) is not a factor of f(x).

### Using the Factor Theorem to Factor Polynomials

At this point, we have very clear strategies for factoring quadratic functions. These are functions where the degree is 2. What happens when we want to factor a polynomial function of a higher degree? It turns out we will be able to use the factor theorem along with some other tools that we will learn later on in this section as a strategy for factoring a polynomial with a degree of 3 or higher. Since we don't have all the tools we need just yet, we will use a given zero to factor. Let's look at some examples.Example #3: Factor each completely. $$f(x) = x^3 + 3x^2 - 49x + 45$$ $$f(-9) = 0$$ Here we are given the fact that f(-9) = 0, which means that [x - (-9)] or (x + 9) is a factor of f(x). Let's set up a synthetic division. $$\frac{x^3 + 3x^2 - 49x + 45}{x + 9} = x^2 - 6x + 5$$ Now, we just need to factor x

^{2}- 6x + 5, which we already know how to do. What are two integers that sum to -6 and have a product of +5? This would be -1 and -5. $$x^2 - 6x + 5 = (x - 1)(x - 5)$$ Putting everything together gives us the following: $$f(x) = x^3 + 3x^2 - 49x + 45$$ $$=(x + 9)(x^2 - 6x +5)$$ $$=(x + 9)(x - 1)(x - 5)$$ Example #4: Factor each completely. $$f(x) = 2x^3 + 11x^2 - 52x - 96$$ $$f(-8) = 0$$ Here we are given the fact that f(-8) = 0, which means that [x - (-8)] or (x + 8) is a factor of f(x). Let's set up a synthetic division. $$\frac{ 2x^3 + 11x^2 - 52x - 96}{x + 8} = 2x^2 - 5x - 12$$ Now, we just need to factor 2x^{2}- 5x - 12, which we already know how to do. We need to find two integers whose product is -24 and whose sum is -5. Two such integers would be -8 and +3, let's use these to write a four-term polynomial. $$2x^2 - 8x + 3x - 12$$ We can now factor this polynomial using grouping. $$2x(x - 4) + 3(x - 4) = (2x + 3)(x - 4)$$ Putting everything together gives us the following: $$f(x) = 2x^3 + 11x^2 - 52x - 96$$ $$=(x + 8)(2x^2 - 5x - 12)$$ $$=(x + 8)(2x + 3)(x - 4)$$#### Skills Check:

Example #1

Determine if g(x) is a factor of f(x). $$f(x)=x^4 - 21x^2 - 100$$ $$g(x)=x - 5$$

Please choose the best answer.

A

Yes

B

No

Example #2

Determine if g(x) is a factor of f(x). $$f(x)=x^4 - 3x^3 + 64x - 192$$ $$g(x)=x - 7$$

Please choose the best answer.

A

Yes

B

No

Example #3

Factor completely. $$f(x)=x^3 - 7x^2 - x + 7$$ $$f(1) = 0$$

Please choose the best answer.

A

$$f(x) = (x - 1)(x + 3)(x + 5)$$

B

$$f(x) = (x + 1)(x + 3)(x + 5)$$

C

$$f(x) = (2x - 3)(x - 3)(x - 1)$$

D

$$f(x) = (2x - 1)(x - 9)(x + 1)$$

E

$$f(x) = (x - 7)(x - 1)(x + 1)$$

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