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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Lesson Settings & Tools
| | 11 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Factor Linear Expressions
Slide of 11
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
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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.
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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.
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. |
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Is This a Monomial?
Determine whether the given expression is a monomial.
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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 |
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Is This a Linear Expression?
Determine whether the given monomial is a linear expression.
Example
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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.
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.
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b Is the expression a linear expression?
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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.
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.
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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.
1
Find the GCF of the Linear Expression
expand_more 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
expand_more 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
expand_more 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.
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Fruit for Sale
After grabbing the snacks for his get-together, Diego saw that the store had a sale on fruit.
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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.
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.
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Find the GCF of a Linear Expression
Consider the given linear expression and identify greatest common factor (GCF) of the terms.
Closure
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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)
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.
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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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Factor Linear Expressions
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