### Monomial Factors of Polynomials

In the last few sections, we learned how to multiply polynomials by using the Distributive Property. All the terms in one polynomial had to be multiplied by all the terms in the other polynomial. In this section, you’ll start learning how to do this process in reverse. The reverse of distribution is called **factoring**.

The total area of the figure above can be found in two ways.

We could find the areas of all the small rectangles and add them: \begin{align*}ab+ac+ad+ae+2a\end{align*}.

Or, we could find the area of the big rectangle all at once. Its width is \begin{align*}a\end{align*} and its length is \begin{align*}b+c+d+e+2\end{align*}, so its area is \begin{align*}a(b+c+d+e+2)\end{align*}.

Since the area of the rectangle is the same no matter what method we use, those two expressions must be equal.

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

To turn the right-hand side of this equation into the left-hand side, we would use the distributive property. To turn the left-hand side into the right-hand side, we would need to **factor** it. Since polynomials can be multiplied just like numbers, they can also be factored just like numbers—and we’ll see later how this can help us solve problems.

**Find the Greatest Common Monomial Factor**

You will be learning several factoring methods in the next few sections. In most cases, factoring takes several steps to complete because we want to **factor completely**. That means that we factor until we can’t factor any more.

Let’s start with the simplest step: finding the greatest monomial factor. When we want to factor, we always look for common monomials first. Consider the following polynomial, written in expanded form:

\begin{align*}ax+bx+cx+dx\end{align*}

A common factor is any factor that appears in all terms of the polynomial; it can be a number, a variable or a combination of numbers and variables. Notice that in our example, the factor \begin{align*}x\end{align*} appears in all terms, so it is a common factor.

To factor out the \begin{align*}x\end{align*}, we write it outside a set of parentheses. Inside the parentheses, we write what’s left when we divide each term by \begin{align*}x\end{align*}:

\begin{align*}x(a+b+c+d)\end{align*}

Let’s look at more examples.

#### Factoring

Factor:

a) \begin{align*}2x+8\end{align*}

We see that the factor 2 divides evenly into both terms: \begin{align*}2x + 8 = 2(x) + 2(4)\end{align*}

We factor out the 2 by writing it in front of a parenthesis: \begin{align*}2( \ )\end{align*}

Inside the parenthesis we write what is left of each term when we divide by 2: \begin{align*}2(x + 4)\end{align*}

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

We see that the factor of 5 divides evenly into all terms: \begin{align*}15x-25= 5(3x)-5(5)\end{align*}

Factor out the 5 to get: \begin{align*}5(3x-5)\end{align*}

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

We see that the factor of 3 divides evenly into all terms: \begin{align*}3a + 9b + 6 = 3(a) + 3(3b) + 3(2)\end{align*}

Factor 3 to get: \begin{align*}3(a + 3b + 2)\end{align*}

#### Finding the Greatest Common Factor

a) \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 \begin{align*}a\end{align*} raised to a different power. The greatest common factor of all the terms is simply \begin{align*}a\end{align*}.

So first we rewrite \begin{align*}a^3-3a^2+4a\end{align*} as \begin{align*}a(a^2) + a(-3a) + a(4)\end{align*}.

Then we factor out the \begin{align*}a\end{align*} to get \begin{align*}a(a^2 - 3a + 4).\end{align*}

b) \begin{align*}12a^4-5a^3+7a^2\end{align*}

The factor \begin{align*}a\end{align*} appears in all the terms, and it’s always raised to at least the second power. So the greatest common factor of all the terms is \begin{align*}a^2\end{align*}.

We rewrite the expression \begin{align*}12a^4-5a^3+7a^2\end{align*} as \begin{align*}(12a^2 \cdot a^2) - (5a \cdot a^2) + (7 \cdot a^2)\end{align*}

Factor out the \begin{align*}a^2\end{align*} to get \begin{align*}a^2(12a^2 - 5a + 7)\end{align*}.

#### Complete Factoring

Factor completely:

a) \begin{align*}3ax+9a\end{align*}

Both terms have a common factor of 3, but they also have a common factor of \begin{align*}a\end{align*}. It’s simplest to factor these both out at once, which gives us \begin{align*}3a(x+3)\end{align*}.

b) \begin{align*}x^3y+xy\end{align*}

Both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are common factors. When we factor them both out at once, we get \begin{align*}xy(x^2+1)\end{align*}.

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

The common factors are 5, \begin{align*}x\end{align*}, and \begin{align*}y\end{align*}. Factoring out \begin{align*}5xy\end{align*} gives us \begin{align*}5xy(x^2-3xy+5xy^2)\end{align*}.

### Example

#### Example 1

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 out the greatest common factor in the following polynomials.

- \begin{align*}2x^2 - 5x\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*}3y + 6z\end{align*}
- \begin{align*}10a^3 - 4ab\end{align*}
- \begin{align*}45y^{12} + 30y^{10}\end{align*}
- \begin{align*}16xy^2 z + 4x^3 y\end{align*}
- \begin{align*}2a - 4a^2 + 6\end{align*}
- \begin{align*}5xy^2 - 10xy + 5y^2\end{align*}

### Review (Answers)

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