9.14: 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 realworld applications like this one that involve factoring polynomial equations.
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CK12 Foundation: 0914S Solving RealWorld Problems By Factoring
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
Use the Pythagorean Theorem:
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
Check:
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
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
Check:
Example C
A rectangle has sides of length
Solution
Make a sketch of this situation:
Using the formula Area = length
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
Check:
Watch this video for help with the Examples above.
CK12 Foundation: Solving RealWorld Problems by Factoring
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:
a⋅b=0⇒a=0 or b=0.
 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
Solution
Make a sketch of this situation:
Using the formula Area = length
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
Check:
Practice
Solve the following application problems:
 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.
 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.
 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?
 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?
 The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.
 A rectangle has a 50foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?
 Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?
 Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?
 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?
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Here you'll learn how to apply polynomial factoring techniques to solve realworld applications involving polynomial equations.