Mastering Factoring Linear Expressions: Greatest Common Factor and Distributive Property

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

Expression

Why It Is a Monomial

5

Any constant is a valid monomial. By the Zero Exponent Property,

5 x^{0} = 5 .

0

The coefficient of a monomial can be

0 .

- 2 x^{5}

The coefficient can be negative.

\frac{x ^{3} y}{5}

A monomial can have numbers in the denominator.

Expression	Why It Is Not a Monomial
$2 x^{- 1}$	The variables of a monomial cannot have negative integer exponents.
$4 \frac{x ^{3}}{y}$	Monomials cannot have variables in the denominator.
$5 x^{3} y^{\frac{1}{2}}$	The variables of a monomial must only have whole number exponents.

Expression

Why It Is Not a Monomial

2 x^{- 1}

The variables of a monomial cannot have negative integer exponents.

4 \frac{x ^{3}}{y}

Monomials cannot have variables in the denominator.

5 x^{3} y^{\frac{1}{2}}

The variables of a monomial must only have whole number exponents.

Monomial	Degree
$3 x$	$1$
$x^{2}$	$2$
$9 x^{3}$	$3$
$x^{3} y$	$4$
$7$	$0$
$\frac{a ^{3} b ^{4} c ^{5}}{1 3}$	$12$
$0$	undefined

Monomial

Degree

3 x

1

x^{2}

2

9 x^{3}

3

x^{3} y

4

7

0

\frac{a ^{3} b ^{4} c ^{5}}{1 3}

12

0

undefined

A student tried to factor the greatest common factor (GCF) out of an algebraic expression. However, they got a wrong answer.

In which step did the student make the mistake?

Write the correct answer by factoring out the GCF of the expression.

Let's start by taking a look at the given expression. 16x+36 We want to factor out the greatest common factor. Start by rewriting each term as a product of its factors. 16x &= 2 * 2 * 2 * 2 * x 36 &= 2 * 2 * 3 * 3 Notice that both terms have 2 as a factor two times. This means the GCF is 2* 2=4. The first step is correct! Next, rewrite each term as a product that involves the GCF. 16x &= 4 * 4x 36 &= 4 * 9 This lets us rewrite the original expression as follows. 16x+36 = 4 * 4x + 4 * 9 This step is also correct! Finally, factor out the GCF. 16x+36 = 4 * (4x+9) This is the step where the student made the mistake. They wrote 36 instead of 9 after factoring out the GCF. This means the answer is Step III.

We already found the correct answer by factoring out the GCF in Part A. Let's take a look at it once more. 4(4x+9)

Consider the following expression.

6 a + 18

Which of the following options are a factored form of the expression? Select all that apply.

Let's begin by taking a look at the original expression. 6a+18 We want to know which of the options from the pool are correct factorizations of the above expression. We can verify each one by expanding the products using the Distributive Property. Let's start with the first option.

The first option resulted in the original expression. This means it is a correct factorization. We can expand the rest of the options in a similar way.

Factored Form	Use the Distributive Property	Simplify
2(3a+9)	2* 3a + 2 * 9	6a+18
2(6a+3)	2* 6a + 2 * 3	12a+6
4(2a+14)	4 * 2a + 4 * 14	8a+56
6(a+3)	6 * a + 6 * 3	6a+18
3(2a+6)	3 * 2a + 3* 6	6a+18

The correct factorizations are 2(3a+9), 6(a+3), and 3(2a+6).

Consider the following expression.

24 x + 40

Which of the following options is not a factored form of the expression?

Let's begin by taking a look at the original expression. 24x+40 We want to know which of the options from the pool is not a correct factorization of the above expression. We can verify each one by expanding the products using the Distributive Property. Let's start with the first option.

The first option resulted in the original expression, meaning that it is a correct factorization. We can expand the rest of the options in a similar way.

Factored Form	Use the Distributive Property	Simplify
2(12x+20)	2* 12x + 2 * 20	24x+40
4(6x+10)	4* 6x + 4 * 10	24x+40
6(4x+6)	6 * 4x + 6 * 6	24x+36
8(3x+5)	8 * 3x + 8 * 5	24x+40

The incorrect factorization is 6(4x+6).

Factor Linear Expressions

Catch-Up and Review

Using the Distributive Property

Monomial

Is This a Monomial?

Degree of a Monomial

Is This a Linear Expression?

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Hint

Solution

Factoring a Linear Expression by GCF

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Hint

Solution

Find the GCF of a Linear Expression

Factoring a Linear Expression Using GCF

Factor Linear Expressions

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