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# Solving Problems by Factoring

## Factor and use the zero product rule

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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 x=\begin{align*}x =\end{align*} the length of the short leg of the triangle; then the other leg will measure x+3\begin{align*}x + 3\end{align*}.

Use the Pythagorean Theorem: a2+b2=c2\begin{align*}a^2+b^2=c^2\end{align*}, where a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*} are the lengths of the legs and c\begin{align*}c\end{align*} is the length of the hypotenuse. When we substitute the values from the diagram, we get x2+(x+3)2=152\begin{align*}x^2+(x+3)^2=15^2\end{align*}.

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.”

x2+x2+6x+92x2+6x+92x2+6x216=225=225=0

Factor out the common monomial: 2(x2+3x108)=0\begin{align*}2(x^2+3x-108)=0\end{align*}

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:

108108=129=12(9)andand12+9=312+(9)=3(Correct choice)

We factor the expression as 2(x9)(x+12)=0\begin{align*}2(x-9)(x+12)=0\end{align*}.

Set each term equal to zero and solve:

x9=0x=9orx+12=0x=12

It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be x=9\begin{align*}x = 9\end{align*}. That means the short leg is 9 feet and the long leg is 12 feet.

Check: 92+122=81+144=225=152\begin{align*}9^2+12^2=81+144=225=15^2\end{align*}, 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 x=\begin{align*}x =\end{align*} one of the numbers; then x+4\begin{align*}x + 4\end{align*} is the other number.

The product of these two numbers is 60, so we can write the equation x(x+4)=60\begin{align*}x(x+4)=60\end{align*}.

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

x2+4xx2+4x60=60=0

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

606060606060=415=4(15)=512=5(12)=610=6(10)andandandandandand4+15=114+(15)=115+12=75+(12)=76+10=4(Correct choice)6+(10)=4

The expression factors as (x+10)(x6)=0\begin{align*}(x+10)(x-6)=0\end{align*}.

Set each term equal to zero and solve:

Since we are looking for positive numbers, the answer must be \begin{align*}x = 6\end{align*}. One number is 6, and the other number is 10.

Check: \begin{align*}6 \cdot 10 = 60\end{align*}, so the answer checks.

#### Example C

A rectangle has sides of length \begin{align*}x + 5\end{align*} and \begin{align*}x - 3\end{align*}. What is \begin{align*}x\end{align*} if the area of the rectangle is 48?

Solution

Make a sketch of this situation:

Using the formula Area = length \begin{align*}\times\end{align*} width, we have \begin{align*}(x+5)(x-3)=48\end{align*}.

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 \begin{align*}(x+9)(x-7)=0\end{align*}.

Set each term equal to zero and solve:

Since we are looking for positive numbers the answer must be \begin{align*}x = 7\end{align*}. So the width is \begin{align*}x - 3 = 4\end{align*} and the length is \begin{align*}x + 5 = 12\end{align*}.

Check: \begin{align*}4 \cdot 12 = 48\end{align*}, 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: \begin{align*}a\cdot b=0 \Rightarrow a=0 \text{ or } b=0.\end{align*}
• 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 \begin{align*}x + 5\end{align*} and \begin{align*}x - 3\end{align*}. What is \begin{align*}x\end{align*} if the area of the rectangle is now 20?

Solution

Make a sketch of this situation:

Using the formula Area = length \begin{align*}\times\end{align*} width, we have \begin{align*}(x+5)(x-3)=20\end{align*}.

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 \begin{align*}(x+7)(x-5)=0\end{align*}.

Set each term equal to zero and solve:

Since we are looking for positive numbers the answer must be \begin{align*}x = 5\end{align*}. So the width is \begin{align*}x - 3 = 2\end{align*} and the length is \begin{align*}x + 5 = 10\end{align*}.

Check: \begin{align*}2 \cdot 10 = 20\end{align*}, 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 \begin{align*}x + 2\end{align*} and \begin{align*}x - 1\end{align*}. What value of \begin{align*}x\end{align*} gives an area of 108?
4. A rectangle has sides of \begin{align*}x - 1\end{align*} and \begin{align*}x + 1\end{align*}. What value of \begin{align*}x\end{align*} 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?