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Chapter 6—Expressions and Equations

6-5 Removing Parentheses

 When you complete the work for this section, you should be able to: Use the distributive property of multiplication and addition to remove parentheses from equations. Simplify equations by removing parentheses and combining like terms.

When you see an algebraic expression in a science or business textbook, it is usually presented in its simplest form. Compare these two versions of the same expression:

2x + 3(x + 4)

and

5x + 12

These expressions are the same. In other words, 2x + 3(x + 4) = 5x + 12. You can demonstrate this fact by substituting any value for the x's and solving the results. You can see that the expression, 5x + 2 is clearly simpler than the version that includes parentheses.

The purpose of this lesson is to demonstrate how to simplify expressions by removing the parentheses. This procedure frequently uses the distributive property.

 Recall The distributive property shows the relationship between the product of one term times the sum of two terms, a(b + c). a(b + c) = ab + ac where a, b, and c are real numbers.

Removing Parentheses from Expressions of Form a(bx + c)

Removing the parentheses from an expression of the form a(bx + c) is a matter of applying the basic distributive property:

Examples A

1. 2(y + 6) = 2y + 12
2. 4(2z + 6) = 8z + 24
3. 1(x + 2) = x + 2

The procedure for removing parentheses require a bit more care when negative values and subtraction are involved.

Examples B

1. 2(y – 6) = 2y – 12
2. 4(3 – P) = 12 – 4P
3. – 3(z + 2) = – 3z – 6
4. – 4(y –  1) = – 4y + 4

Examples & Exercises

 Removing Parentheses Apply the distributive property to eliminate the parentheses from these expressions.

Removing Sets of Parentheses that are Added/Subtracted

Consider this messy looking expression: 2(x + 1) + 3(x + 6)

Working from left to right:

2(x + 1) + 3(x + 6) = 2x + 2 + 3x + 18

Collecting like terms:

2x + 2 + 3x + 18 = 5x + 20

Result:

2(x + 1) + 3(x + 6) = 5x + 20

Examples C

Examples & Exercises

 Removing Parentheses Use these interactive Examples & Exercises to strengthen your understanding and build your skills:

Removing Sets of Parentheses that are Multiplied

Tutorial Example

Consider this  expression: (2 + 4)(3 + 5)

Here is the simplest approach to reducing the expression:

Sum the terms enclosed in parentheses:

(2 + 4)(3 + 5) = (6)(8)

Do the multiplication:

(6)(8) = 48

Solution: (2 + 4)(3 + 5) = 48

(a + b)(c + d) = ac + ad + bc + bd

(a + b)(c + d) = –(ac + ad + bc + bd) = ac ad bc bd

(a + b)(c + d + e) = ac + ad + ae + bc + bd + be

(a + b)(c + d)(e + f) = (ac + ad + bc + bd)(e + f) = ace + acf  + ade + + adf + bce + bcf + bde + bdf

Tutorial Example

Reduce this expression: (4x - 2)(5x + 1)

(4x - 2)(5x + 1) = 20x2 + 1 -10x -2

Examples C

(2 + 4)(3 + 5)

(a + b)(c + d)

a(c + d) + b(c + d)

ac + ad + bc + bd

(2 + 4)(3 + 5)

2(3 + 5) + 4(3 + 5)

6 + 10 + 12 + 20 = 48

(2 + 4)(3 + 5) = 48

Note: This is the same as (2 + 4)(3 + 5) = (6)(8) = 48

(3 6)(2 + 5)

3(2 + 5) 6(2 + 5)

6 + 15 – 12 – 30 =

(3 6)(2 + 5) = – 21

(2x + 1)(4x 1)

8x2 2x + 4x 1

8x2 + 2x 1 Step 1: Multiply the first term in the first expression by the first term in the second expression
Step 2:Multiply the first term in the first expression by the second term in the second expression
Step 3. Multiply the second term in the first expression by the first term in  the second expression
Step 4.Multiply the second term in the first expression by the second term in the second expression
Step 5 Sum the results
Step 6 Collect like terms

Result: (2x + 4)(5x - 3) = 10x2 + 14x - 12

 NoteThis is one of those operations in math that is actually quite simple to perform, but often awkward to explain. Basically, you multiply both of the two  terms in the first parentheses by both of the two  terms in the second  parentheses.

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