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Expanding expressions

Here you will learn how to expand expressions before simplifying the resulting algebraic expressions.

Students will first learn about expanding expressions as part of expressions and equations in 6th grade.

What is expanding expressions?

Expanding expressions (or multiplying out) is the process by which you use the distributive property to remove parentheses from an algebraic expression.

To do this, you need to multiply out the parentheses by multiplying everything outside of the parentheses by everything inside the parentheses. Then, if needed, you simplify the resulting expression by combining the like terms.

For example,

2(x+5) = 2x+10

For this example,

The expression has two sets of parentheses. You will need to simplify by combining like terms after expanding the expression.

\begin{aligned} & 2(x+5)+3(x-1) \\\\ & =2 x+10+3 x-3 \\\\ & =5 x+7 \end{aligned}

What is expanding expressions?

What is expanding expressions?

Common Core State Standards

How does this relate to 6th grade math?

  • Grade 6 – Expressions and Equations (6.EE.A.3)
    Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 \, (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 \, (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y .

How to expand expressions

In order to expand expressions:

  1. Use the distributive property to “multiply out” the parentheses in the expression.
  2. Combine the like terms.

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREE
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[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

[FREE] End of Year Math Assessments (Grade 4 & Grade 5)

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREE

Expanding expressions examples

Example 1: one set of parentheses

Expand:

9 \, (y+6)

  1. Use the distributive property to “multiply out” the parentheses in the expression.

\begin{aligned} & 9 \, (y+6) \\\\ & =(9 \times y)+(9 \times 6) \\\\ & =9 y+54 \end{aligned}

2Combine the like terms.

There are no like terms in the expression, so you are finished.

The expanded expression is 9y+54 .

Example 2: one set of parentheses

Expand and simplify:

6 \, (𝑡 + 8)–12

\begin{aligned} & 6 \, (t+8)-12 \\\\ & =(6 \times t)+(6 \times 8)-12 \\\\ & =6 t+48-12 \end{aligned}

6 t+48-12


= 6t+36


So the expanded and simplified expression is 6t+36 .

Example 3: one set of parentheses

5 b+7(b-2)-18

\begin{aligned} & 5 b+7(b-2)-18 \\\\ & =5 b+(7 \times b)-(7 \times 2)-18 \\\\ & =5 b+7 b-14-18 \end{aligned}

\begin{aligned} & 5 b+7 b-14-18 \\\\ & =12 b-32 \end{aligned}


So the expanded and simplified expression is 12b-32 .

Example 4: two sets of parentheses

Expand and simplify:

8(x-2)+6(x+3)

Multiply out the first set of parentheses.


\begin{aligned} & 8 \, (x-2) \\\\ & =(8 \times x)-(8 \times 2) \\\\ & =8 x-16 \end{aligned}


The first set of parentheses expands to 8x–16 .


\begin{aligned} & 6 \, (x+3) \\\\ & =(6 \times x)+(6 \times 3) \\\\ & =6 x+18 \end{aligned}


The second set of parentheses expands to 6x + 18 .


The expression is now,


8 x-16+6 x+18

Expanding Expressions example 4


First, use the commutative property to group the like terms together.


Underline the two x terms (8x and +6x) and combine them. Then do the same for the two constants (-16 and +18). Remember to underline the sign in front of the number too!


8 x+6 x=14 x


-16 and +18 = 2


(Note that -16 + 18 can also be written as 18-16 ).


So the expanded and simplified expression is 14x + 2 .

Example 5: constants and variables outside of the parentheses

Expand and simplify:

2 x(x+6)-3(x-2)

Multiply out the first set of parentheses.


\begin{aligned} & 2 x \, (x+6) \\\\ & =(2 x \times x)+(2 x \times 6) \\\\ & =2 x^2+12 x \end{aligned}


The first set of parentheses expands to 2x^2 + 12x .


Multiply out the second set of parentheses.


\begin{aligned} & -3 \, (x-2) \\\\ & =(-3 \times x)+(-3 \times 2) \end{aligned}


*Note: since the minus sign stays with the 3 to become negative 3, you will now add the products.


=-3 x+6


The second set of parentheses expands to –3x + 6 .


The expression is now,


2 x^2+12 x-3 x+6

The first term will be 2x^2 since there are no other exponents. Underline the two x terms (12x and -3x) and combine them.


Remember to underline the sign in front of the number too. Since there is only one constant, the expression will end in +6.


12 x-3 x=9 x


The expanded and simplified expression is 2x^2 + 9x + 6 .

Example 6: variables in both terms in the parentheses

Expand and simplify:

3 \, (2 x-6 y)-5 \, (x-2 y)

Multiply the first set of parentheses. Another method you can use is to write the terms in an area model:


Expanding Expressions example 6 image 1


(A positive number times a negative number equals a negative number so 3 \times -6y gives a negative answer. You need to write -18y ).


The first set of parentheses expands to 6x–18y .


Multiply out the second set of parentheses. Remember, you are multiplying both x and -2y by -5 .


Expanding Expressions example 6 image 2


(A negative number times a negative number equals a positive number so -5 \times -2y gives a positive answer. You need to write +10y ).


The second set of parentheses expands to –5x + 10y .


The expression is now,


6 x-18 y-5 x+10 y

Underline the two x terms (6x and -5x) and the two y terms (-18y and +10y).


Remember to underline the sign in front of the number too.


Expanding Expressions example 6 image 3


The expanded and simplified expression is x-8y .

Teaching tips for expanding expressions

  • Assess students’ prior knowledge before beginning. Review key vocabulary terms including algebraic expression, parentheses, simplify, exponents, distributive property, binomial, polynomial, terms, coefficients. Review adding and subtracting negative numbers as needed.

  • Allowing learners to use templates or organizers when practicing this skill or completing worksheets, such as the area model shown above, will help them organize their work as they complete the process step-by-step.

  • Be sure students have had plenty of practice expanding expressions with one set of parentheses before moving on to expanding expressions with binomials or polynomials.

  • The expanding expression tool kit you will provide students is essential for higher levels of mathematics.

Easy mistakes to make

  • Multiplying all terms in the set of parentheses
    You must multiply the term outside the parentheses by every term inside the parentheses.
    For example,

    2\left(6 x^2-3 x\right)=12 x^2-3 x

    Here, you have not multiplied the value outside of the parentheses by the second term. The correct answer is:

    2\left(6 x^2-3 x\right)=12 x^2-6 x

  • Squaring a term
    When you square something, you multiply it by itself.
    For example,

    \begin{aligned} 3^{2}&=3\times 3\\ x^{2}&=x\times x\\ (5y)^{2}&=5y\times 5y \end{aligned}

  • Combining like terms
    When you combine like terms, you must include the sign in front of the number. Be sure to underline the sign along with the number when combining the like terms.
    For example,

    Expanding Expressions image 1

Practice expanding expressions questions

1. Expand and simplify:

 

5 \, (m+9)-4 m

5m+9
GCSE Quiz False

m+9
GCSE Quiz False

9m+45
GCSE Quiz False

m+45
GCSE Quiz True

Expand each set of parentheses:

 

5m + 45-4m

 

Combine like terms:

 

m+45

2. Expand and simplify:

 

4 h-2(h+8)-1

2h-17
GCSE Quiz True

2h+17
GCSE Quiz False

2h-5
GCSE Quiz False

h-15
GCSE Quiz False

Expand each set of parentheses:

 

4h-2h-16-1

 

Combine like terms:

 

2h-17

3. Expand and simplify:

 

3 \, (x+7)-2 \, (x+3)

5x+15
GCSE Quiz False

x+15
GCSE Quiz True

x+27
GCSE Quiz False

5x+27
GCSE Quiz False

Expand each set of parentheses:

 

3x + 21-2x-6

 

Combine like terms:

 

x+15

4. Expand and simplify:

 

8 \, (y-5)+5 \, (y-2)

13y-30
GCSE Quiz False

3y-50
GCSE Quiz False

13y-50
GCSE Quiz True

3y-30
GCSE Quiz False

Expand each set of parentheses:

 

8y-40 + 5y-10

 

Combine like terms:

 

13y-50

5. Expand and simplify:

 

5 x \, (3 x-2)-4 x \, (2 x+3)

23x^2-22x
GCSE Quiz False

7x^2-5
GCSE Quiz False

7x^2-22x
GCSE Quiz True

7x^2-2x
GCSE Quiz False

Expand each set of parentheses:

 

15x^{2}-10x-8x^{2}-12x

 

Combine like terms:

 

7x^{2}-22x

6. Expand and simplify:

 

5 \, (6 x-2 y)-2 \, (8 x-5 y)

14x
GCSE Quiz True

14x-20y
GCSE Quiz False

46x
GCSE Quiz False

14x+20y
GCSE Quiz False

Expand each set of parentheses:

 

30x-10y-16x+10y

 

Combine like terms:

 

14x

Expanding expressions FAQs

How do you expand an expression?

To expand an expression, you need to multiply out the parentheses by multiplying everything outside of the parentheses by everything inside of the parentheses. Then, if needed, you simplify the resulting expression by combining the like terms.

How do you combine like terms?

Combine like terms by first identifying the like terms, then adding them together using the value of their coefficients.

Why is expanding an expression an important skill to have?

Expanding expressions are useful when solving and graphing quadratic equations, rational expressions, and solving non-linear simultaneous equations.

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