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9.6: Monomial Factors of Polynomials

Difficulty Level: Basic Created by: CK-12
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Suppose that the height of a golf ball above the ground is represented by the expression 2x2+20x\begin{align*}-2x^2 + 20x\end{align*}, where x\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? In this Concept, you'll learn how to find monomial factors of polynomials like the one representing the height of the golf ball in this example.

Guidance

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. In this Concept, you will start learning how to do this process using a different method called factoring.

Factoring: Take the factors that are common to all the terms in a polynomial. Then multiply the common factors by a parenthetical expression containing all the terms that are left over when you divide out the common factors.

Let’s look at the areas of the rectangles again: Area=length×width\begin{align*}Area = length \times width\end{align*}. The total area of the figure on the right can be found in two ways.

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

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

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

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

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

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

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

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

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

The answers are the same no matter which method you use:

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

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.

Example A

Factor:

1. 15x25\begin{align*}15x-25\end{align*}
2. 3a+9b+6\begin{align*}3a+9b+6\end{align*}

Solution:

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

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

2. 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 use examples where different powers can be factored and there is more than one common factor.

Example B

Find the greatest common factor.

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

Solution:

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*}.

Example C

Find the greatest common factor.

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

Solution:

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*}.

Guided Practice

Find the greatest common factor.

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

Solution:

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*}

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: Polynomial Equations in Factored Form (9:29)

Factor the common factor from the following polynomials.

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

Mixed Review

1. Rewrite in standard form: \begin{align*}-4x+11x^4-6x^7+1-3x^2\end{align*}. State the polynomial's degree and leading coefficient.
2. Simplify \begin{align*}(9a^2-8a+11a^3)-(3a^2+14a^5-12a)+(9-3a^5-13a)\end{align*}.
3. Multiply \begin{align*}\frac{1}{3} a^3\end{align*} by \begin{align*}(36a^4+6)\end{align*}.
4. 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.
1. Write the equation to represent this situation.
2. Graph using its intercepts.
3. Give three possible combinations to make this sentence true.
1. Explain how to use mental math to simplify 8(12.99).

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Color Highlighted Text Notes

Vocabulary Language: English Spanish

common factor

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

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.

Monomial

A monomial is an expression made up of only one term.

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

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