Suppose that the height of a golf ball above the ground is represented by the expression \begin{align*}-2x^2 + 20x\end{align*}, where \begin{align*}x\end{align*} is the horizontal displacement of the ball. What is the greatest common factor of the terms in this expression? Could you use the greatest common factor to factor the expression?

### Factoring

We have been multiplying polynomials by using the Distributive Property, where all the terms in one polynomial must be multiplied by all terms in the other polynomial. Now, you will start learning how to do this process using a different method called factoring.

**Factoring **is a technique that takes the factors that are common to all the terms in a polynomial out of the expression. The final form of a factored expression has all common factors multiplied by a parenthetical expression containing all the terms that are left over after you divide out the common factors.

#### Let's find the total area of the following figure:

The area of each of the smaller rectangles is described by the formula: \begin{align*}Area = length \times width\end{align*}. The total area of the figure on the above can be found in two ways:

**Method 1:** Find the areas of all the small rectangles and add them.

Blue rectangle \begin{align*}= ab\end{align*}

Orange rectangle \begin{align*}= ac\end{align*}

Red rectangle \begin{align*}=ad\end{align*}

Green rectangle \begin{align*}=ae\end{align*}

Purple rectangle \begin{align*}=2a\end{align*}

Total area \begin{align*}=ab+ac+ad+ae+2a\end{align*}

**Method 2:** Find the area of the big rectangle all at once.

\begin{align*}\text{Length} & = a\\ \text{Width} & = b+c+d+e+2\\ \text{Area} & = a(b+c+d+e+2)\end{align*}

Note that the answers are the same no matter which method you use:

\begin{align*}ab+ac+ad+ae+2a=a(b+c+d+e+2)\end{align*}

The left side of the equation shows the unfactored form while the right side shows the factored form of the same expression.

#### Finding the Greatest Common Monomial Factor

Once we get a polynomial in factored form, it is easier to solve the polynomial equation. But first, we need to learn how to factor. Factoring can take several steps because we want to factor completely until we cannot factor any more.

A **common factor** can be a number, a variable, or a combination of numbers and variables that appear in every term of the polynomial.

When a common factor is factored from a polynomial, you divide each term by the common factor. What is left over remains in parentheses.

#### Let's factor the following expressions:

- \begin{align*}15x-25\end{align*}

We see that the factor of 5 divides evenly from all terms.

\begin{align*}15x-25=5(3x-5)\end{align*}

- \begin{align*}3a+9b+6\end{align*}

We see that the factor of 3 divides evenly from all terms.

\begin{align*}3a+9b+6=3(a+3b+2)\end{align*}

Now we will complete some problems where different powers can be factored and there is more than one common factor.

#### Let's find the greatest common factor for the following expressions:

- \begin{align*}a^3-3a^2+4a\end{align*}

Notice that the factor \begin{align*}a\end{align*} appears in all terms of \begin{align*}a^3-3a^2+4a\end{align*} but each term has a different power of \begin{align*}a\end{align*}. The common factor is the lowest power that appears in the expression. In this case the factor is \begin{align*}a\end{align*}.

Let’s rewrite: \begin{align*}a^3-3a^2+4a=a(a^2)+a(-3a)+a(4)\end{align*}

Factor \begin{align*}a\end{align*} to get \begin{align*}a(a^2-3a+4)\end{align*}.

- \begin{align*}5x^3y-15x^2y^2+25xy^3\end{align*}

The common factor is \begin{align*}5xy\end{align*}.

When we factor \begin{align*}5xy\end{align*}, we obtain \begin{align*}5xy(x^2-3xy+5y^2)\end{align*}.

### Examples

#### Example 1

Earlier, you were told that the height of a golf ball above the ground is represented by the expression \begin{align*}-2x^2 + 20x\end{align*}, where \begin{align*}x\end{align*} is the horizontal displacement of the ball. What is the greatest common factor of the terms in this expression? What is the factored form of the expression?

Notice that -2 divides evenly into both of the coefficients. We can factor out a -2:

\begin{align*}-2x^2 + 20x &= -2(x^2)-2(-10x)\\ &=-2(x^2 -10x)\end{align*}Note that you could factor out 2 instead of -2 and the result would still be correct.

Now, notice that the factor \begin{align*}x\end{align*} appears in all of the terms. We can factor out \begin{align*}x\end{align*}:

\begin{align*}-2(x^2+10x)&=-2(x\cdot x-10\cdot x) \\ &=-2x(x-10)\end{align*}

The factored form of this equation is \begin{align*}-2x(x-10)\end{align*} and the greatest common factor of the terms is \begin{align*}-2x\end{align*}.

#### Example 2

Find the greatest common factor.

\begin{align*}16x^2y^3z^2+4x^3yz+8x^2y^4z^5\end{align*}

First, look at the coefficients to see if they share any common factors. They do: 4.

Next, look for the lowest power of each variable, because that is the most you can factor out. The lowest power of \begin{align*}x\end{align*} is \begin{align*}x^2\end{align*}. The lowest powers of \begin{align*}y\end{align*} and \begin{align*}z\end{align*} are to the first power.

This means we can factor out \begin{align*}4x^2yz\end{align*}. Now, we have to determine what is left in each term after we factor out \begin{align*}4x^2yz\end{align*}:

\begin{align*}16x^2y^3z^2+4x^3yz+8x^2y^4z^5=4x^2yz(4y^2z+x+2y^3z^4)\end{align*}

### Review

Factor the common factor from the following polynomials.

- \begin{align*}36a^2+9a^3-6a^7\end{align*}
- \begin{align*}yx^3 y^2+12x+16y\end{align*}
- \begin{align*}3x^3-21x\end{align*}
- \begin{align*}5x^6+15x^4\end{align*}
- \begin{align*}4x^3+10x^2-2x\end{align*}
- \begin{align*}-10x^6+12x^5-4x^4\end{align*}
- \begin{align*}12xy+24xy^2+36xy^3\end{align*}
- \begin{align*}5a^3-7a\end{align*}
- \begin{align*}45y^{12}+30y^{10}\end{align*}
- \begin{align*}16xy^2z+4x^3y\end{align*}

**Mixed Review**

- Rewrite in standard form: \begin{align*}-4x+11x^4-6x^7+1-3x^2\end{align*}. State the polynomial's degree and leading coefficient.
- Simplify \begin{align*}(9a^2-8a+11a^3)-(3a^2+14a^5-12a)+(9-3a^5-13a)\end{align*}.
- Multiply \begin{align*}\frac{1}{3} a^3\end{align*} by \begin{align*}(36a^4+6)\end{align*}.
- Melissa made a trail mix by combining \begin{align*}x\end{align*} ounces of a 40% cashew mixture with \begin{align*}y\end{align*}ounces of a 30% cashew mixture. The result is 12 ounces of cashews.
- Write the equation to represent this situation.
- Graph using its intercepts.
- Give three possible combinations to make this sentence true.

- Explain how to use mental math to simplify 8(12.99).

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.6.