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Pre-Algebra View details

7. Factor Linear Expressions

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Pre-Algebra /

Chapter 4

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.

Student Learning Objectives:
Identify monomials and their degree Use the greatest common factor to factor a linear expression

	11 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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Linear expressions can be used to describe many real-life situations. In this lesson, we will learn how to factor linear expressions and use factoring to make expressions simpler.

Catch-Up and Review

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

Algebraic Expressions
Linear Expression
Distributive Property
Greatest Common Factor

Challenge

Using the Distributive Property

In the following applet, different algebraic expressions are expanded using the Distributive Property.

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. A monomial is the product of powers of variables and a constant called the coefficient.

A single-term expression is a monomial only if all of its variables have whole numbers as exponents. However, 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.

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. 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.

Example

Using a Coupon For Snacks

Diego has a coupon to a local store.

This coupon gives him 5 % off when buying snacks! If x represents the total cost of his snacks, then 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.

The following expression represents how much Diego will pay for his snacks after using the coupon.

0.95x Is this a monomial?

Is the expression a linear expression?

Hint

Identify the coefficient and the variable of the expression.

What is the power of the variable shown in the expression?

Solution

After using the coupon, Diego will pay 0.95 x. Here, x is a variable that represents the original price, and 0.95 is a coefficient that represents the portion of the price Diego will pay.

Coefficient Variable [-8mu] ↖ ↗ [-10mu] 0.95 x The only operation in this expression is multiplication between the variable and the coefficient, so the expression is a monomial.

We already know that the given expression is a monomial. Now, let's see if it is also a linear expression. Remember, when no exponent is shown, the exponent is 1.

0.95x = 0.95x^1 There is only one variable, and its exponent is 1, so it is a linear expression. We can think of this monomial as a linear expression with no constant term. Coefficient Variable [-8mu] ↖ ↗ [-10mu] 0.95 x_(Linear Term)

Discussion

Factoring a Linear Expression by GCF

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

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.

Find the GCF of the Linear Expression

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.

Rewrite Each Term in Terms of the GCF

Next, rewrite each term of the initial expression as the product of the GCF and another factor. 12x &= 4* 3x 4 &= 4* 1 Rewrite the initial expression with these products. 12x+4 ⇕ [0.25em] 4* 3x+ 4* 1

Factor Out the GCF

Finally, use the Distributive Property to factor out the GCF. 4* 3x+ 4* 1 ⇕ [0.25em] 4( 3x + 1 ) Consider the expression inside the parentheses to determine whether it is possible to continue the factorization. These terms do not have any common factors, so the linear expression is completely factored.

Any common factor can be used to factor a linear expression, though the most common and efficient way is by using the GCF.

Example

Fruit for Sale

A store has a sale on fresh fruit.

Mark bought 9 bags of apples and 3 bunches of bananas. He uses a to represent apples and b to represent bananas and writes the following expression. 9a+3b Mark wants to divide this fruit into 3 equal groups. Rewrite the expression by factoring out 3.

Hint

Find the common factors of each term.

Solution

To factor out 3 from the expression representing the fruit Mark got at the grocery store, we start by writing the factors of each term. Both 9a and 3b have 3 as a factor.

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)

The expression can be written as 3( 3a+ b). This means the fruit can be split into 3 equal groups, with each group containing 3a apples and b bananas.

Pop Quiz

Find the GCF of a Linear Expression

Identify greatest common factor (GCF) of the terms.

Closure

Factoring a Linear Expression Using GCF

Consider the linear expression presented at the beginning of the lesson. 18x+15 Begin by finding the factors of 18x and 15. 18x &= 2 * 3 * 3 * x 15 &= 3 * 5 They only share one factor, 3, so 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 3.1

Diego's mother likes knitting as a stress-relieving pastime. When Diego went to the supermarket, he thought that selling some of her mother's crafts would be a great idea to make some extra money.

Of course, Diego is willing to give his mother her cut. She receives x for each scarf or pair of socks sold, and Diego receives the remaining amount. Diego sells 4 items for a total of 4x+16 dollars. How much does Diego receive for each item sold?

We are told that the following expression represents the sale of four items crafted by Diego's mother. 4x+16 Since 4 items were sold, we will factor 4 out from the above expression. Start by writing each term as a product involving 4. 4x &= 4 * x 16 &= 4 * 4 We can rewrite the original expression as follows. 4x+16 = 4 * x + 4 * 4 We can now factor out 4 from the above expression. 4x+16=4(x+4) This means that each item sold gets Diego x+4 dollars. Since x represents his mother's cut, the remaining term corresponds to Diego's cut. This means that Diego makes $ 4 for every item sold.

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