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7. Factor Linear Expressions
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Chapter 4
7. 

Factor Linear Expressions

This lesson focuses on the techniques for factoring linear expressions, a fundamental skill in algebra. It explains how to use the greatest common factor (GCF) and the distributive property to simplify expressions. These methods are not just theoretical; they have practical applications too. For example, understanding how to factor linear expressions can help in solving problems related to finance, engineering, and even grocery shopping. The lesson aims to make these mathematical concepts accessible and useful in everyday life, helping both students and professionals make more informed decisions.
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Factor Linear Expressions
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There are several real-life scenarios that can be modeled using linear expressions. This type of expression is crucial for the development of more advanced topics. This lesson will discuss how to factor linear expressions and how to take advantage of this technique.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Using the Distributive Property

In the following applet, different algebraic expressions are expanded using the Distributive Property.
Distributive Property Applet
Is it possible to work these expressions in reverse? Consider the following example. 18x+15 Try rewriting the above expression as the product of 3 and a different expression.
Discussion

Monomial

A monomial is an algebraic expression consisting of only one term. It is a product of powers of variables and a constant called the coefficient.

The monomial 2x^2y^3 with a coefficient of 2, and variables x and y.

A single-term expression is a monomial only if all of its variables have whole numbers — non-negative and integers — as exponents. However, variables with positive exponents in the denominator are excluded because they are equivalent to a power in the numerator with the opposite exponent, according to the Quotient of Powers Property. Consider the following example. 5x/y^2 = 5xy^(- 2) In other words, if a variable in the denominator of an expression, it is not a monomial. The following are valid examples of monomials.

Expression Why It Is a Monomial
5 Any constant is a valid monomial. By the Zero Exponent Property, 5x^0=5.
0 The coefficient of a monomial can be 0.
- 2x^5 The coefficient can be negative.
x^3y/5 A monomial can have numbers in the denominator.

Although they appear to be monomials at first glance, the single-term expressions in the following table do not satisfy the definition of a monomial.

Expression Why It Is Not a Monomial
2x^(- 1) The variables of a monomial cannot have negative integer exponents.
4x^3/y Monomials cannot have variables in the denominator.
5 x^3y^(12) The variables of a monomial must only have whole number exponents.
Pop Quiz

Is This a Monomial?

Determine whether the given expression is a monomial.

Determining if an expression is a monomial random generator
Discussion

Degree of a Monomial

The degree of a monomial is the sum of the exponents of its variable factors. If a variable has no exponent written in it, it is assumed to be 1. Additionally, all nonzero constants have a degree of 0. The constant 0 does not have a degree.

Monomial Degree
3x 1
x^2 2
9x^3 3
x^3y 4
7 0
a^3b^4c^5/13 12
0 undefined
This lesson will work with monomials of degree 1, which are also known as linear expressions.
Pop Quiz

Is This a Linear Expression?

Determine whether the given monomial is a linear expression.

Determining if an expression is a monomial random generator
Example

Using a Coupon For Snacks

Diego is having friends over this evening to watch movies and hang out. It would be great if they had some snacks while watching the movies! He suddenly remembers that he has a coupon in his wallet.

Coupon For Five Percent Off on Snacks

This coupon gives him 5 % off when buying snacks! Diego uses x to represent the total cost of the snacks he will buy. When he uses the coupon, 5 % of the sales price will be discounted from this total. x-0.05x=0.95x This means that Diego will only pay 95 % of the total.

a The following expression represents how much Diego will pay for his snacks after using the coupon.
0.95x Is this a monomial?
b Is the expression a linear expression?

Hint
a Identify the coefficient and the variable of the expression.
b What is the power of the variable shown in the expression?

Solution
a Begin by taking a look at the expression that represents how much Diego will pay after using the coupon.

0.95x The variable is x and the coefficient is 0.95. The variable x represents the normal price of the snacks Diego wants to buy. The coefficient 0.95 represents the total after a 5 % discount.

Coefficient and Variable

The only operation involved in this expression is the multiplication of the variable and the coefficient, so this expression is a monomial.

b The given expression has already been determined to be a monomial. Now it is asked whether it is a linear expression. Recall that when no exponent is written on a variable, the exponent is assumed to be 1. 0.95x = 0.95x^1 There is only one variable in the above expression and its exponent is one, so it is a linear expression. Think of this monomial as a linear expression that has no constant term.
Coefficient and Variable of a Linear Term
Discussion

Factoring a Linear Expression by GCF

A monomial whose degree is 1 is a linear expression. Another linear expression can be created by either adding or subtracting a constant from a monomial of degree 1.

A diagram illustrating the components of a linear expression in the form ax+b, including the linear term, its coefficient and variable, and the constant term. and variable, and the constant term.
When the linear term and the constant term of a linear expression share common factors, it is possible to factor out their greatest common factor (GCF) using the Distributive Property. Consider the following linear expression. 12x+4 This expression can be factored by GCF by following these steps.
1
Find the GCF of the Linear Expression
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Start by finding the GCF of the linear expression. Rewrite each term as the product of its factors. 12x &= 2* 2 * 3 * x 4 &= 2* 2 The GCF of the initial expression is 2* 2= 4.

2
Rewrite Each Term in Terms of the GCF
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Next, rewrite each term of the initial expression as the product of the GCF and another factor. 12x &= 4* 3x 4 &= 4* 1 Now write the initial expression as follows. 12x+4 ⇕ [0.25em] 4* 3x+ 4* 1

3
Factor Out the GCF
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Finally, use the Distributive Property to factor out the GCF. 4* 3x+ 4* 1 ⇕ [0.25em] 4( 3x + 1 ) The expression between the parentheses can to be examined to determine whether it is possible to continue the factorization. Here, the terms do not have any more common factors, so the linear expression is completely factored.

Please note that any common factor can be used to factor a linear expression. The most common way of doing it is by using the GCF.
Example

Fruit for Sale

After grabbing the snacks for his get-together, Diego saw that the store had a sale on fruit.

Fruit on Sale Ad
He got kind of carried away and bought 9 apples and 3 bananas. However, if he does not eat the fruit soon, it will spoil. Diego decides that sharing the fruit with his friends would be good. He uses a to represent apples and b to represent bananas and writes the following expression. 9a+3b Diego will split the fruit into three equal parts for him and his two friends. Rewrite the above expression by factoring out 3.

Hint

Find the common factors of each term.

Solution
The following expression represents the fruit that Diego got at the grocery store. 9a+3b Diego wants to split the fruit into three equal parts. This should not be a hard task since both 9a and 3b share the factor 3. Begin by writing the factors of each term.
9a+3b
Rewrite

Rewrite 9a as 3* 3a

3* 3a+3b
Rewrite

Rewrite 3b as 3* b

3* 3a + 3* b
FactorOut

Factor out 3

3(3a+b)
This means that the expression can be rewritten as 3(3a+b). The factor 3 means that the fruit will be split into 3 equal groups. The factor 3a+b means that each person will get 3 apples and 1 banana.
Pop Quiz

Find the GCF of a Linear Expression

Consider the given linear expression and identify greatest common factor (GCF) of the terms.

Find the GCF of a linear expression
Closure

Factoring a Linear Expression Using GCF

The challenge presented at the start of the lesson can be solved by using the methods learned in this chapter. Consider the given linear expression. 18x+15 Begin by finding the factors of 18x and 15. 18x &= 2 * 3 * 3 * x 15 &= 3 * 5 Notice that they only share one factor, 3. This means their greatest common factor (GCF) is 3. Next, rewrite each term as a product involving the GCF. 18x &= 3 * 6x 15 &= 3 * 5 Finally, rewrite the expression by factoring out 3.

18x+15 &= 3* 6x + 3* 5 &= 3(6x+5)


Level 1
Level 2
Level 3
Factor Linear Expressions
Exercise 1.1

Factor out the indicated whole number from the given expression.

Factor out 3 from the following expression. 3x+6
Factor out 5 from the following expression. 15y+25
Factor out 2 from the following expression. 16z+2

We are asked to factor out 3 from the following expression. 3x+6 We can start by rewriting each term as a product that includes 3 as a factor. 3x &= 3 * x 6 &= 3 * 2 Next, rewrite the expression accordingly. 3x+6 = 3* x + 3 * 2 Finally, we can factor out 3 as requested. 3* x + 3 * 2 = 3 (x+2)


Let's take a look at the next expression. 15y+25 This time we are asked to factor out 5 from the expression. Begin by writing each term as a product that includes 5 as a factor. 15y &= 5 * 3y 25 &= 5 * 5 Use this to rewrite the original expression. 15y+25 = 5* 3y + 5 * 5 Finish by factoring 5 out from the expression. 5* 3y + 5 * 5 = 5(3y+5)


Let's take a look at the last expression. 16z+2 This time we want to factor out 2 from the expression. Rewrite each term as a product involving 2 as a factor. 16z &= 2 * 8z 2 &= 2 * 1 Rewrite the original expression using the factored form of each term. 16z+2 = 2* 8z + 2 * 1 Finish by factoring 2 out of the expression. 2 * 8z + 2 * 1 = 2 (8z+1)

E
Hint & Answer
S
Solution

Factor out the indicated integer from each given expression.

Factor out 7 from the following expression. 14a-7
Factor out 4 from the following expression. 16b-20
Factor out 3 from the following expression. - 9c + 6

We are asked to factor out 7 from the following expression. 14a-7 We will start by rewriting each term as a product that includes 7 as a factor. 14a &= 7 * 2a 7 &= 7 * 1 Next, rewrite the expression accordingly. Keep in mind that the original expression is a subtraction expression. 14a-7 = 7* 2a - 7 * 1 Finally, we can factor out 7, as requested. 7* 2a - 7 * 1 = 7 (2a-1)


Let's take a look at the next expression. 16b-20 This time we are asked to factor out 4 from the expression. Begin by writing each term as a product that includes 4 as a factor. 16b &= 4 * 4b 20 &= 4 * 5 Use this to rewrite the original expression. 16b-20 = 4* 4b - 4 * 5 Finish by factoring 4 out from the expression. 4* 4b - 4 * 5 = 4(4b-5)


Let's take a look at the last expression. -9 c+6 This time we want to factor out 3 from the expression. Rewrite each term as a product involving 3 as a factor. -9c &= 3 * (- 3c) 6 &= 3 * 2 Rewrite the original expression using the factored form of each term. - 9c+6 = 3* (- 3c) + 3 * 2 Finish by factoring 3 out of the expression. 3 * (- 3c) + 3 * 6 = 3 (- 3c+2)

E
Hint & Answer
S
Solution

Factor out the indicated rational number from each given expression.

Factor out 12 from the following expression. 3/2p+5/2
Factor out 13 from the following expression. 2q+2/3

We are asked to factor out 12 from the following expression. 3/2p+5/2 Let's start by rewriting each term as a product that includes 12 as a factor. 3/2p &= 1/2 * 3p [0.8em] 5/2 &= 1/2 * 5 Next, rewrite the expression accordingly. 3/2p+5/2 = 1/2 * 3p + 1/2 * 5 Finally, we can factor out 12, as requested. 3/2p+5/2 = 1/2 (3p+5)


Let's take a look at the next expression. 2q+2/3 This time we are asked to factor out 13 from the expression. However, notice that the term 2q has no 3 in the denominator. Let's try rewriting 2q to make it easier to factor out 13.

Now we can rewrite the original expression so that both terms have 3 in the denominator. 2q+2/3 = 6q/3 + 2/3 We can now rewrite each term as a product that includes 13 as a factor. 6q/3 &= 1/3 * 6q [0.8em] 2/3 &= 1/3 * 2 Use this to rewrite the sum. 6q/3+2/3 = 1/3 * 6q + 1/3 * 2 Finish by factoring 13 out from the expression. 1/3 * 6q + 1/3 * 2 = 1/3(6q+2)

E
Hint & Answer
S
Solution

Factor out the greatest common factor from the given expression.

20x+50
21y+14

Let's take a look at the given expression. 20x+50 We are asked to factor out the greatest common factor (GCF) of the expression. Let's start by finding the factors of each term. 20x &= 2 * 2 * 5 * x 50 &= 2 * 5 * 5 These terms share the factors 2 and 5, so the GCF is 2 * 5= 10. We can now rewrite each term as a product that involves the GCF. 20x &= 10 * 2x 50 &= 10 * 5 This lets us rewrite the original expression. 20x+50= 10 * 2x + 1 * 5 Finally, we can factor out the GCF. 10 * 2x + 1 * 5 = 10(2x+5)


Let's look at the next expression. 21y+14 We also want to factor out the GCF from the expression. Start by writing the factors of each term. 21y &= 3y * 7 14 &= 2 * 7 This time, the terms only share one common factor, 7. This is the GCF. We can now rewrite the expression like we did in Part A. 21y+14 = 7 * 3y + 7 * 2 We finish by factoring out the GCF. 7 * 3y + 7 * 2 = 7(3y+2)

E
Hint & Answer
S
Solution

Determine whether the expression given in each case is a monomial.

3x^2
5/y^3
3z/4

Let's take a look at the given expression. 3x^2 The variable is x and the coefficient is 3. The exponent of the variable is 2. 3 x^2 Since the variable is not in a denominator and its exponent is a positive integer, the expression is a monomial.


Let's take a look at the next expression. 5/y^3 The variable is y and the coefficient is 5. The exponent of the variable is 3. 5/y^3 However, the variable is in the denominator of the fraction, so the expression is not a monomial.


Let's take a look at the last expression. 3z/4 Before we identify the coefficient and the variable of this expression, let's rewrite it. 3z/4 = 3/4 * z We can see that the variable is z and the coefficient is 34. Since there is no exponent written in the variable, it is assumed to be 1. 3/4 z^1 Since the variable is not in the denominator and its exponent is a positive integer, the expression is a monomial.

E
Hint & Answer
S
Solution

Determine whether the monomial given in each case is a linear expression.

5a^2
b
1/2c

Let's take a look at the given monomial. 5a^2 We can see that the variable is a and its exponent is 2. 5 a^2 Since the exponent of the variable is not 1, this monomial is not a linear expression.


Let's take a look at the next monomial. b This monomial is just the variable b. When no exponent is written on a variable, it is assumed to be 1. b^1 There is only one variable in this monomial and its exponent is 1, which means that this monomial is a linear expression.


Let's take a look at the last monomial. 1/2c This time the variable is c. Once again, there is no exponent written on it, so it is assumed to be 1. 1/2 c^1 This monomial also has only one variable and its exponent is 1, so the monomial is a linear expression.

E
Hint & Answer
S
Solution

Factor Linear Expressions

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