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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

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. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3(6x+5)"]}}

Discussion

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.$y_{2}5x =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. |

$5x_{3}y $ | 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. |

$4yx_{3} $ | Monomials cannot have variables in the denominator. |

$5x_{3}y_{21}$ | The variables of a monomial must only have whole number exponents. |

Pop Quiz

Determine whether the given expression is a monomial.

Discussion

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 |
---|---|

$3x$ | $1$ |

$x_{2}$ | $2$ |

$9x_{3}$ | $3$ |

$x_{3}y$ | $4$ |

$7$ | $0$ |

$13a_{3}b_{4}c_{5} $ | $12$ |

$0$ | undefined |

Pop Quiz

Determine whether the given monomial is a linear expression.

Example

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.

$0.95x $

Is this a monomial? {"type":"choice","form":{"alts":["Yes","No"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}

b Is the expression a linear expression?

{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

a Identify the coefficient and the variable of the expression.

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

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 $1.$ **is** a linear expression. Think of this monomial as a linear expression that has no constant term.

$0.95x=0.95x_{1} $

There is only one variable in the above expression and its exponent is one, so it Discussion

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 $1.$

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

Start by finding the GCF of the linear expression. Rewrite each term as the product of its factors.

$12x4 =2⋅2⋅3⋅x=2⋅2 $

The GCF of the initial expression is $2⋅2=4.$ 2

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.

$12x4 =4⋅3x=4⋅1 $

Now write the initial expression as follows.
$12x+4⇕4⋅3x+4⋅1 $

3

Factor Out the GCF

Finally, use the Distributive Property to factor out the GCF.

$4⋅3x+4⋅1⇕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. Example

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

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 $3.$ {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["a","b"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3(3a+b)"]}}

Find the common factors of each term.

The following expression represents the fruit that Diego got at the grocery store.
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.

$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 $3$

$3(3a+b)$

Pop Quiz

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

Closure

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 $15.$
$18x15 =2⋅3⋅3⋅x=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.
$18x15 =3⋅6x=3⋅5 $

Finally, rewrite the expression by factoring out $3.$ $18x+15 =3⋅6x+3⋅5=3(6x+5) $

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