Suppose that the area of a poster hanging on your wall is 1000 square inches and that the height is 15 more inches than the width. How would you go about finding the height and width of the poster? What equation would you set up? How could you use factoring to solve the equation? In this Concept, you'll learn how to factor an expression completely in order to solve real-world problems such as this one.

### Guidance

We say that a polynomial is **factored completely** when we factor as much as we can and we are unable to factor any more. Here are some suggestions that you should follow to make sure that you factor completely.

\begin{align*}\checkmark\end{align*}

\begin{align*}\checkmark\end{align*}

\begin{align*}\checkmark\end{align*}

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#### Example A

*Factor the following polynomials completely.*

(a) \begin{align*}2x^2-8\end{align*}

(b) \begin{align*}x^3+6x^2+9x\end{align*}

**Solution:**

(a) Look for the common monomial factor: \begin{align*}2x^2-8=2(x^2-4)\end{align*}

(b) Recognize this as a perfect square and factor as \begin{align*}x(x+3)^2\end{align*}

**Factoring Common Binomials**

The first step in the factoring process is often factoring the common monomials from a polynomial. Sometimes polynomials have common terms that are binomials. For example, consider the following expression:

\begin{align*}x(3x+2)-5(3x+2)\end{align*}

You can see that the term \begin{align*}(3x+2)\end{align*}

\begin{align*}(3x+2)(x-5)\end{align*}

This expression is now completely factored. Let’s look at some more examples.

#### Example B

*Factor* \begin{align*}3x(x-1)+4(x-1)\end{align*}

**Solution:** \begin{align*}3x(x-1)+4(x-1)\end{align*}

When we factor the common binomial, we get \begin{align*}(x-1)(3x+4)\end{align*}

**Solving Real-World Problems Using Polynomial Equations**

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

#### Example C

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

**Solution:** \begin{align*}x=\end{align*}

\begin{align*}x(x+4)=60\end{align*}

Write the polynomial in standard form.

\begin{align*}x^2+4x&=60\\
x^2+4x-60&=0\end{align*}

**Factor:** \begin{align*}-60=6 \times(-10)\end{align*}*and* \begin{align*}6+(-10)=-4\end{align*}

\begin{align*}-60=-6 \times 10\end{align*}*and* \begin{align*}-6+10=4\end{align*}

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

**Solve:**

\begin{align*}x+10=0 && x-6& =0\\
\text{or} \\
x=-10 && x& =6\end{align*}

Since we are looking for positive numbers, the answer must be positive.

\begin{align*}x=6\end{align*}

**Check:** \begin{align*}6 \cdot 10=60\end{align*}

### Guided Practice

*Factor completely: \begin{align*}24x^3-28x^2+8x\end{align*} 24x3−28x2+8x.*

**Solution:**

First, notice that each term has \begin{align*}4x\end{align*}

\begin{align*}24x^3-28x^2+8x=4x(6x^2-7x+2)\end{align*}

Next, factor the trinomial in the parenthesis. Since \begin{align*}a\neq 1\end{align*}

\begin{align*} 12&=-1\cdot -12 && and && -1+-12=-13\\
12&=-2\cdot -6 && and && -2+-6=-8\\
12&=-3\cdot -4 && and && -3+-4=-7\end{align*}

Rewrite the trinomial using \begin{align*}-7x=-3x-4x\end{align*}

\begin{align*}6x^2-7x+2=6x^2-3x-4x+2=3x(2x-1)-2(2x-1)=(3x-2)(2x-1)\end{align*}

The final factored answer is:

\begin{align*}4x(3x-2)(2x-1)\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Factor by Grouping and Factoring Completely (13:57)

Factor completely.

- \begin{align*}2x^2+16x+30\end{align*}
- \begin{align*}12c^2-75\end{align*}
- \begin{align*}-x^3+17x^2-70x\end{align*}
- \begin{align*}6x^2-600\end{align*}
- \begin{align*}-5t^2-20t-20\end{align*}
- \begin{align*}6x^2+18x-24\end{align*}
- \begin{align*}-n^2+10n-21\end{align*}
- \begin{align*}2a^2-14a-16\end{align*}
- \begin{align*}2x^2-512\end{align*}
- \begin{align*}12x^3+12x^2+3x\end{align*}

Solve the following application problems.

- One leg of a right triangle is seven feet longer than the other leg. The hypotenuse is 13 feet. 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?
- The product of two positive numbers is 120. Find the two numbers if one number is seven more than the other.
- Framing Warehouse offers a picture-framing service. The cost for framing a picture is made up of two parts. The cost of glass is $1 per square foot. The cost of the frame is $2 per linear foot. If the frame is a square, what size picture can you get framed for $20.00?

**Mixed Review**

- The area of a square varies directly with its side length.
- Write the general variation equation to model this sentence.
- If the area is 16 square feet when the side length is 4 feet, find the area when \begin{align*}s=1.5 \ feet\end{align*}.

- The
**surface area**is the total amount of surface of a three-dimensional figure. The formula for the surface area of a cylinder is \begin{align*}SA=2 \pi r^2+2 \pi rh\end{align*}, where \begin{align*}r=radius\end{align*} and \begin{align*}h=height \ of \ the \ cylinder\end{align*}. Determine the surface area of a soup can with a radius of 2 inches and a height of 5.5 inches. - Factor \begin{align*}25g^2-36\end{align*}. Solve this polynomial when it equals zero.
- What is the greatest common factor of \begin{align*}343r^3 t, 21t^4\end{align*}, and \begin{align*}63rt^5\end{align*}?
- Discounts to the hockey game are given to groups with more than 12 people.
- Graph this solution on a number line.
- What is the domain of this situation?
- Will a church group with 12 members receive a discount?