1 Topic 2.1.1 Simplifying Algebraic Expressions Simplifying Algebraic Expressions

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Topic 2.1.1Topic 2.1.1

Simplifying Algebraic

Expressions

Simplifying Algebraic

Expressions

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Topic2.1.1

Simplifying Algebraic ExpressionsSimplifying Algebraic Expressions

California Standard:4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12.

What it means for you:You’ll combine like terms to simplify expressions.

Key words:• algebraic expression• simplify• like terms

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Unless you’re told otherwise, you always need to give algebraic solutions in the simplest form possible.

For example:

3x + 2 – 7 – x can be written as: 2x – 5

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Algebraic Expressions Contain Variables

Algebraic expressions are made up of terms.

A term can be a product of numbers and variables, like 4x2 or 5x, or just a number.

For example, the algebraic expression 4x2 – 5x + 7 – 2x2 + 2x – 3 has six terms.

The terms are separated by plus and minus signs.

Each sign “belongs” to the term that it’s in front of.

8x2 + 2xy – 6y + 7y2 – 2

Invisible + sign. This minus sign belongs to the 6y term.

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Guided Practice

Solution follows…

Write the number of terms in the expressions in Exercises 1–4.

1. 3x2 + 4x – 2

2. 8x4 + 7x3 + 2x2 – 8

3. 3x2

4. 8 + 7xy – 2xy9 + 4x7 + 3y2 + 55y3

5. Which variable is multiplied by –4 in the algebraic expression 4x2 – 4y + 8 + 4xy?

6. Counting from left to right, which term is the fourth term in the algebraic expression 8x2 + 2xy – 6y + 9xy3 – 4?

3

4

1

6

y

9xy3

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Like Terms Can Be Combined

Like terms are terms with identical variables that have identical exponents.

The terms 4x2 and –2x2 are like terms because they have the same variable, x, with the same exponent, 2.

–5x and 2x are like terms, and 7 and –3 are like terms too.

Like terms can be combined using the distributive, commutative, and associative properties.

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Example 1

Solution follows…

Simplify 4x2 – 2x2.

Solution

4x2 – 2x2

= (4 – 2)x2

= 2x2

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Combine Like Terms to Simplify Expressions

To simplify an algebraic expression, use number properties to first group and then combine like terms.

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Example 2

Solution follows…

Simplify the following:

Solution

a)

a) (4x – 5) – 2x

(4x – 5) – 2x

4x + (–5 – 2x)=

=

=

=

=

4x + (–2x – 5)

(4x + –2x) – 5

[(4 – 2)x] – 5

2x – 5

Associative property of addition

Commutative property of addition

Associative property of addition

Distributive property

Example continues…

b) 5x2 – 3x + 7x2 – 4x + 9

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Topic2.1.1


Example 2

Simplify the following:

Solution

b)

b) 5x2 – 3x + 7x2 – 4x + 9

5x2 – 3x + 7x2 – 4x + 9

5x2 + 7x2 – 3x – 4x + 9=

=

=

=

=

(5x2 + 7x2) + (–3x – 4x) + 9

(5 + 7)x2 + (–3 – 4)x + 9

12x2 + (–7)x + 9

12x2 – 7x + 9

a) (4x – 5) – 2x

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Simplify the following expressions in Exercises 7–14.

7. 7a + 12a – 4 8. 7a + 3b – 5a

9. 9x + (20 – 5x) 10. 5 – 10x – 2x – 7

11. 7x2 + 7 + 20x2 + 3x 12. 3 – 8x2 + 4x2 + 6x2 – 10

13. 5a – 4 × 3a + 7 × 2 – 3 × 6

14. 3 × 4a – 2 × 5a2 + 2 × 2a2

Topic2.1.1


Guided Practice

Solution follows…

19a – 4 2a + 3b

4x + 20 –12x – 2

27x2 + 3x + 7 2x2 – 7

–7a – 4

–6a2 + 12a

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In Exercises 15–18, simplify the expressions and determine the number of terms in each simplified form.

15. (4a – 9) + (2a – 18)

16. 15n + 3n + 8 – 2 – 6

17. 6a2 + 3a – 9a2 + 2a + 7 + 6

18. 6a + 3 × 7b – 2 × 5c + 7 – 9 + 2 × 4c

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Guided Practice

Solution follows…

6a – 27; 2 terms

18n; 1 term

–3a2 + 5a + 13; 3 terms

6a + 21b – 2c – 2; 4 terms

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Independent Practice

Solution follows…

In Exercises 1–5, determine the number of terms in each algebraic expression:

1. 7b + 14a – 4 2. 2a

3. (27x2 + 4x) – 13 4. 5 +10x + 20x2 + 3a

5. 2x + 4xy + 4x2 – (10 + 12y + 19y2)

3 1

3 4

6

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Solution follows…

In Exercises 6–9, simplify each algebraic expression:

6. 12m – 7 + 3c – 7m – 8c

7. 4a + 3b + 11a – 8b

8. 2 × 6x – 3 × 5x – 3x × 4 + 5 × 2x + 12x

9. 5m × 3 + 2 × 7m – 4m × 4 + 7 × 2m – 17m

5m – 5c – 7

15a – 5b

7x

10m

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In Exercises 10–13, simplify each algebraic expression:

10. 3x + y – x

11. a – b + b – a

12. (17.8n + 13.08q) – 3.8q – 9.9n

13. 0.4x2 + x – x2 – 0.14 –

1

2

1

2

1

8

3

10

8

10

6

12

3

8

1

2

5

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Solution follows…

7.9n + 9.28q

1

22 x + y

1

2

3

8– a + b

1

2

3

8–0.1x2 + x – 0.765

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Solution follows…

14. Which expression below simplifies to 2x + 1?

i. x + 4x + 4 – 3 – 3x + x

ii. 7 + 5x – 6 + 5 – 4x + x – 5

15. Which expression below simplifies to 3x?

i. 5 + 4x2 – 3x – 2 + 4x2 – 8x2 – 3

ii. –2x2 + 6 + 4x2 – 3 + 3x – 2x2 – 3

ii) simplifies to 2x + 1

ii) simplifies to 3x

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Solution follows…

In Exercises 16–17, find a simplified expression for the perimeter of the figure.

16.

17.

12x + 2

4a + 4x + 10

4x + 1

2x

3

2x

7

2x

2a2a

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18. Juan bought 3 baseball cards for b dollars each and 2 baseball cards for c dollars each. He has bought 4 comic books for $5.00 each.

Write and simplify an algebraic expression showing the total money Juan spent on baseball cards and comic books.

19. Three friends Tom, Leo, and Maria have several pieces of candy to eat. Tom has (2x + 4) pieces of candy, Leo has (8 – 2x) pieces of candy, and Maria has 8 pieces of candy.

Write and simplify an algebraic expression showing the total number of pieces of candy the three friends have to eat.

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Solution follows…

3b + 2c + 20

20

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Round UpRound Up

You’ve combined like terms before, in earlier grades — so this Topic should feel like good practice.

It’s always important to give your final answers to algebraic problems in the simplest form.

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1 Topic 2.1.1 Simplifying Algebraic Expressions Simplifying Algebraic Expressions