The area of a rectangle is . The width of the rectangle is . What is the length?

### Long Division of Polynomials

Even though it does not seem like it, factoring is a form of division. Each factor goes into the larger polynomial evenly, without a remainder.

For example, take the polynomial . If we use factoring by grouping, we find that the factors are . If we multiply these three factors together, we will get the original polynomial. So, if we divide by , we should get

How many times does go into Since it goes in times.

Place above the term in the polynomial.

*Multiply* by both terms in the **divisor** ( and -3) and place them under their like terms. *Subtract* from the **dividend** Pull down the next two terms and repeat.

Since goes into a total of -4 times.

After multiplying both terms in the divisor by -4, place that under the terms you brought down. When subtracting, notice that everything cancels out. Therefore is indeed a factor.

When dividing polynomials, not every divisor will go in evenly to the dividend. If there is a remainder, write it as a fraction over the divisor.

Let's divide the following polynomials using long division.

Set up the problem using a long division bar.

How many times does go into Since it goes in times.

*Multiply* by the divisor. *Subtract* that from the dividend.

Repeat the previous steps. Now, how many times does go into ? It goes in 4 times.

This is the limit of this process. cannot go evenly into because it has a higher degree. Therefore, is a remainder. The complete answer would be

Determine if goes evenly into . If so, try to factor the divisor and quotient further.

First, do the long division. If goes in evenly, then the remainder will be zero.

This means that and both go evenly into . Let’s see if we can factor either or further.

and .

Therefore, You can multiply these to check the work. A binomial with a degree of one is a **factor** of a larger polynomial if it goes evenly into it. In this problem, and are all factors of . This indicates that 1, 1, -3, and are all solutions of

**Factor Theorem:** A polynomial, , has a factor, , if and only if .

In other words, if is a **solution** or a **zero**, then the factor, divides evenly into .

Now, let's determine if 5 is a solution of .

To see if 5 is a solution, we need to divide the factor into . The factor that corresponds with 5 is .

Since there is a remainder, 5 is not a solution.

### Examples

#### Example 1

Earlier, you were asked to find the length of the rectangle.

First, do the long division.

This means that and both go evenly into .

can't be factored further, so it is the rectangle's length.

#### Example 2

Divide: .

Make sure to put a placeholder in for the term.

The final answer is .

#### Example 3

Is a factor of ? If so, find any other factors.

Divide into and if the remainder is zero, it is a factor.

is a factor. Let’s see if factors further. Yes, the factors of -27 that add up to -6 are -9 and 3. Therefore, the factors of are , and .

#### Example 4

What are the real-number solutions to Example 3?

The solutions would be -4, 9, and 3; the opposite sign of each factor.

#### Example 5

Determine if 6 is a solution to .

To see if 6 is a solution, we need to divide into .

Because the remainder is not zero, 6 is not a solution.

### Review

Divide the following polynomials using long division.

Determine all the real-number solutions to the following polynomials, given one factor.

Determine all the real number solutions to the following polynomials, given one zero.

Find the equation of a polynomial with the given zeros.

- 4, -2, and
- 1, 0, and 3
- -5, -1, and
**Challenge**Find*two*polynomials with the zeros 8, 5, 1, and -1.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.9.