<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Solving Problems by Factoring

## Factor and use the zero product rule

%
Progress
Progress
%
Solving Problems by Factoring

What if you had a triangle in which one leg was one unit shorter than the other and the hypotenuse was 5 units? How could you find the length of the two legs? After completing this Concept, you'll be able to solve real-world applications like this one that involve factoring polynomial equations.

### Guidance

Now that we know most of the factoring strategies for quadratic polynomials, we can apply these methods to solving real world problems.

#### Example A

One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the triangle.

Solution

Let the length of the short leg of the triangle; then the other leg will measure .

Use the Pythagorean Theorem: , where and are the lengths of the legs and is the length of the hypotenuse. When we substitute the values from the diagram, we get .

In order to solve this equation, we need to get the polynomial in standard form. We must first distribute, collect like terms and rewrite in the form “polynomial = 0.”

Factor out the common monomial:

To factor the trinomial inside the parentheses, we need two numbers that multiply to -108 and add to 3. It would take a long time to go through all the options, so let’s start by trying some of the bigger factors:

We factor the expression as .

Set each term equal to zero and solve:

It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be . That means the short leg is 9 feet and the long leg is 12 feet.

Check: , so the answer checks.

#### Example B

The product of two positive numbers is 60. Find the two numbers if one number is 4 more than the other.

Solution

Let one of the numbers; then is the other number.

The product of these two numbers is 60, so we can write the equation .

In order to solve we must write the polynomial in standard form. Distribute, collect like terms and rewrite:

Factor by finding two numbers that multiply to -60 and add to 4. List some numbers that multiply to -60:

The expression factors as .

Set each term equal to zero and solve:

Since we are looking for positive numbers, the answer must be . One number is 6, and the other number is 10.

Check: , so the answer checks.

#### Example C

A rectangle has sides of length and . What is if the area of the rectangle is 48?

Solution

Make a sketch of this situation:

Using the formula Area = length width, we have .

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and rewrite:

Factor by finding two numbers that multiply to -63 and add to 2. List some numbers that multiply to -63:

The expression factors as .

Set each term equal to zero and solve:

Since we are looking for positive numbers the answer must be . So the width is and the length is .

Check: , so the answer checks.

Watch this video for help with the Examples above.

### Vocabulary

• Polynomials can be written in expanded form or in factored form. Expanded form means that you have sums and differences of different terms:
• The factored form of a polynomial means it is written as a product of its factors.
• Zero Product Property: The only way a product is zero is if one or more of the terms are equal to zero:
• We say that a polynomial is factored completely when we factor as much as we can and we are unable to factor any more.

### Guided Practice

Consider the rectangle in Example C with sides of length and . What is if the area of the rectangle is now 20?

Solution

Make a sketch of this situation:

Using the formula Area = length width, we have .

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and rewrite:

Factor by finding two numbers that multiply to -35 and add to 2. List some numbers that multiply to -35:

The expression factors as .

Set each term equal to zero and solve:

Since we are looking for positive numbers the answer must be . So the width is and the length is .

Check: , so the answer checks.

### Practice

Solve the following application problems:

1. One leg of a right triangle is 1 feet longer than the other leg. The hypotenuse is 5. Find the dimensions of the right triangle.
2. One leg of a right triangle is 7 feet longer than the other leg. The hypotenuse is 13. Find the dimensions of the right triangle.
3. A rectangle has sides of and . What value of gives an area of 108?
4. A rectangle has sides of and . What value of gives an area of 120?
5. The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.
6. A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?
7. Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?
8. Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?
9. Framing Warehouse offers a picture framing service. The cost for framing a picture is made up of two parts: glass costs $1 per square foot and the frame costs$2 per foot. If the frame has to be a square, what size picture can you get framed for \$20?