3.3: Simplifying Algebraic Expressions

Recall the commutative and associative properties of multiplication.

The Commutative Property of Multiplication. If a and b are any integers, then

a · b = b · a, or equivalently, ab = ba.

The Associative Property of Multiplication. If a, b, and c are any integers, then

(a · b) · c = a · (b · c), or equivalently, (ab)c = a(bc).

The commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer.

Example 1

Simplify: 2(3x).

Solution

Use the associative property to regroup, then simplify.

\[ \begin{aligned} 2(3x) = (2 \cdot 3)x ~ & \textcolor{red}{ \text{ Regrouping with the associative property.}} \\ = 6x ~ & \textcolor{red}{ \text{ Simplify: } 2 \cdot 3 = 6.} \end{aligned}\nonumber \]

Exercise

Simplify: −5(7y)

Answer

−35y

The statement 2(3x)=6x is an identity. That is, the left-hand side and right-hand side of 2(3x)=6x are the same for all values of x. Although the derivation in Example 1 should be the proof of this statement, it helps the intuition to check the validity of the statement for one or two values of x.

If x = 4, then

\[ \begin{array}{c c c} 2(3x) = 2(3( \textcolor{red}{4})) & \text{and} & 6x = 6( \textcolor{red}{4}) \\ = 2(12) & & = 24 \\ = 24 \end{array}\nonumber \]

If x = −5, then

\[ \begin{array}{c c c} 2(3x) = 2(3( \textcolor{red}{-5})) & \text{and} & 6x = 6( \textcolor{red}{-5}) \\ =2(-15) & & = -30 \\ = -30 \end{array}\nonumber \]

The above calculations show that 2(3x)=6x for both x = 4 and x = −5. Indeed, the statement 2(3x)=6x is true, regardless of what is substituted for x.

Example 2

Simplify: (−3t)(−5).

Solution

In essence, we are multiplying three numbers, −3, t, and −5, but the grouping symbols ask us to multiply the −3 and the t first. The associative and commutative properties allow us to change the order and regroup.

\[ \begin{aligned} (-3t)(-5) = ((-3)(-5))t ~ & \textcolor{red}{ \text{ Change the order and regroup.}} \\ = 15t ~ & \textcolor{red}{ \text{ Multiply: } (-3)(-5) = 15.} \end{aligned}\nonumber \]

Exercise

Simplify: (−8a)(5)

Answer

−40a

Example 3

Simplify: (−3x)(−2y)

Solution

In essence, we are multiplying four numbers, −3, x, −2, and y, but the grouping symbols specify a particular order. The associative and commutative properties allow us to change the order and regroup.

\[ \begin{aligned} (-3x)(-2y) =((-3)(-2))(xy) ~ & \textcolor{red}{ \text{ Change the order and regroup.}} \\ = 6xy ~ & \textcolor{red}{ \text{ Multiply: } (-3)(-2)=6.} \end{aligned}\nonumber \]

Exercise

Simplify: (−4a)(5b)

Answer

−20ab

Speeding Things Up

The meaning of the expression 2 · 3 · 4 is clear. Parentheses and order of operations are really not needed, as the commutative and associative properties explain that it doesn’t matter which of the three numbers you multiply together first.

• You can multiply 2 and 3 first:

\[ \begin{aligned} 2 \cdot 3 \cdot 4 &= (2 \cdot 3) \cdot 4 \\ &= 6 \cdot 4 \\ &= 24. \end{aligned}\nonumber \]

• Or you can multiply 3 and 4 first:

\[ \begin{aligned} 2 \cdot 3 \cdot 4 &= 2 \cdot (3 \cdot 4) \\ &= 2 \cdot 12 \\ &= 24. \end{aligned}\nonumber \]

• Or you can multiply 2 and 4 first:

\[ \begin{aligned} 2 \cdot 3 \cdot 4 &= (2 \cdot 4) \cdot 3 \\ &= 8 \cdot 3 \\ &= 24. \end{aligned}\nonumber \]

So, it does not matter which two factors you multiply first.

Of course, this would not be the case if there were a mixture of multiplication and other operators (division, addition, subtraction). Then we would have to strictly follow the “Rules Guiding Order of Operations.” But if the only operator is multiplication, the order of multiplication is irrelevant.

Thus, when we see 2(3x), as in Example 1, we should think “It’s all multiplication and it doesn’t matter which two numbers I multiply together first, so I’ll multiply the 2 and the 3 and get 2(3x)=6x.”

Our comments apply equally well to a product of four or more factors. It simply doesn’t matter how you group the multiplication. So, in the case of (−3x)(−2y), as in Example 3, find the product of −2 and −3 and multiply the result by the product of x and y. That is, (−3x)(−2y)=6xy.

Example 4

Simplify: (2a)(3b)(4c).

Solution

The only operator is multiplication, so we can order and group as we please. So, we’ll take the product of 2, 3, and 4, and multiply the result by the product of a, b, and c. That is,

\[(2a)(3b)(4c)=24abc\nonumber \]

Exercise

Simplify: (−3x)(−2y)(−4z)

Answer

−24xyz.

The Distributive Property

Multiplication is distributive with respect to addition.

The Distributive Property

If a, b, and c are any integers, then

a · (b + c) = a · b + a · c, or equivalently, a(b + c) = ab + ac.

For example, if we follow the “Rules Guiding Order of Operations” and first evaluate the expression inside the parentheses, then

\[ \begin{aligned} 3(4+5)=3(9) ~ & \textcolor{red}{ \text{ Parentheses first: } 4+5 = 9.} \\ =27. ~ & \textcolor{red}{ \text{ Multiply: } 3(9) = 27.} \end{aligned}\nonumber \]

But if we “distribute” the 3, we get the same answer.

\[ \begin{aligned} 3(4+5) =3(4+5) ~ & \begin{aligned} \textcolor{red}{ \text{ Each number in parentheses is multiplied}} \\ \textcolor{red}{ \text{ by the number 3 outside the parentheses.}} \end{aligned} \\ = 3(4) +3(5) ~ \\ = 12 + 15 ~ & \textcolor{red}{ \text{ Multiply first: } 3(4) = 12,~ 3(5) = 15.} \\ =27 ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber \]

Example 5

Use the distributive property to simplify: 3(4x + 5).

Solution

Distribute the 3.

\[ \begin{aligned} 3(4x+5) = 3(4x)+3(5) ~ & \begin{aligned} \textcolor{red}{ \text{ Each number in parentheses is multiplied}} \\ \textcolor{red}{ \text{ by the number 3 outside the parentheses.}} \end{aligned} \\ =12x + 15 ~ & \textcolor{red}{ \text{ Multiply first: } 3(4x)=12x,~ 3(5)=15.} \end{aligned}\nonumber \]

Exercise

Use the distributive property to simplify: 2(5z+7).

Answer

10z + 14

Multiplication is also distributive with respect to subtraction.

The Distributive Property

If a, b, and c are any integers, then

a · (b − c) = a · b − a · c, or equivalently, a(b − c) = ab − ac.

The application of this form of the distributive property is identical to the first, the only difference being the subtraction symbol.

Example 6

Use the distributive property to simplify: 5(3x − 2).

Solution

Distribute the 5.

\[ \begin{aligned} 5(3x - 2) = 5(3x)-5(2) ~ & \begin{aligned} \textcolor{red}{ \text{ Each number in parentheses is multiplied }} \\ \textcolor{red}{ \text{ by the number 5 outside the parentheses.}} \end{aligned} \\ = 15x - 10 ~ & \textcolor{red}{ \text{ Multiply first: } 5(3x) = 15x, 5(2) = 10.} \end{aligned}\nonumber \]

Exercise

Use the distributive property to simplify: 7(4a − 5).

Answer

28a − 35

Example 7

Remove parentheses: (a) −9(2t + 7), and (b) −5(4 − 3y).

Solution

a) Use the distributive property.

\[ \begin{aligned} -9(2t+7) = -9(2t)+(-9)(7) ~ & \textcolor{red}{ \text{ Distribute multiplication by }-9.} \\ = -18t + (-63) ~ & \textcolor{red}{ \text{ Multiply: } -9(2t) = -18t \text{ and } -9(7) = -63.} \\ = - 18t - 63 ~ & \textcolor{red}{ \text{ Write the answer in simpler form.}} \\ ~ & \textcolor{red}{ \text{ Adding } -63 \text{ is the same as subtracting 63.}} \end{aligned}\nonumber \]

b) Use the distributive property.

\[ \begin{aligned} -5(4-3y) = -5(4)-(-5)(3y) ~ & \textcolor{red}{ \text{ Distribute multiplication by }-5.} \\ = -20-(-15y) ~ & \textcolor{red}{ \text{ Multiply: } -5(4) = -20 \text{ and } -5(3y) = -15y.} \\ = - 18t - 63 ~ & \textcolor{red}{ \text{ Write the answer in simpler form.}} \\ ~ & \textcolor{red}{ \text{ Subtracting } -15y \text{ is the same as adding } 15y.} \end{aligned}\nonumber \]

Exercise

Remove parentheses: −3(4t − 11).

Answer

−12t + 33

Writing Mathematics

Example 7 stresses the importance of using as few symbols as possible to write your final answer. Hence, −18t − 63 is favored over −18t + (−63) and −20 + 15y is favored over −20 − (−15y). You should always make these final simplifications.

Moving a Bit Quicker

Once you’ve applied the distributive property to a number of problems, showing all the work as in Example 7, you should try to eliminate some of the steps. For example, consider again Example 7(a). It’s not difficult to apply the distributive property without writing down a single step, getting:

\[−9(2t + 7) = −18t − 63.\nonumber \]

Here’s the thinking behind this technique:

1. First, multiply −9 times 2t, getting −18t.
2. Second, multiply −9 times +7, getting −63.

Note that this provides exactly the same solution found in Example 7(a).

Let try this same technique on Example 7(b).

\[−5(4 − 3y) = −20 + 15y\nonumber \]

Here’s the thinking behind this technique.

1. First, multiply −5 times 4, getting −20.
2. Second, multiply −5 times −3y, getting +15y.

Note that this provides exactly the same solution found in Example 7(b).

Extending the Distributive Property

Suppose that we add an extra term inside the parentheses.

Distributive Property

If a, b, c, and d are any integers, then

a(b + c + d) = ab + ac + ad.

Note that we “distributed” the a times each term inside the parentheses. Indeed, if we added still another term inside the parentheses, we would “distribute” a times that term as well.

Example 8

Remove parentheses: −5(2x − 3y + 8).

Solution

We will use the “quicker” technique, “distributing” −5 times each term in the parentheses mentally.

\[ -5(2x - 3y +8)=-10x + 15y -40\nonumber \]

Here is our thought process:

1. First, multiply −5 times 2x, getting −10x.
2. Second, multiply −5 times −3y, getting +15y.
3. Third, multiply −5 times +8, getting −40.

Exercise

Remove parentheses: −3(4a − 5b + 7)

Answer

−12a + 15b − 21

Example 9

Remove parentheses: −4(−3a + 4b − 5c + 12).

Solution

We will use the “quicker” technique, “distributing” −4 times each term in the parentheses mentally.

\[ -4(-3a + 4b - 5c +12) = 12a - 16b + 20c - 48\nonumber \]

Here is our thought process:

1. First, multiply −4 times −3a, getting 12a.
2. Second, multiply −4 times +4b, getting −16b.
3. Third, multiply −4 times −5c, getting +20c.
4. Fourth, multiply −4 times +12, getting −48.

Exercise

Remove parentheses: −2(−2x + 4y − 5z − 11).

Answer

4x − 8y + 10z + 22

Distributing a Negative

It is helpful to recall that negating is equivalent to multiplying by −1.

Multiplying by −1

Let a be any integer, then

(−1)a = −a and −a = (−1)a.

We can use this fact, combined with the distributive property, to negate a sum.

Example 10

Remove parentheses: −(a + b).

Solution

Change the negative symbol into multiplying by −1, then distribute the −1.

\[ \begin{aligned} -(a + b) =(-1)(a+b) ~ & \textcolor{red}{ \text{ Negating is equivalent to multiplying by } -1.} \\ =-a-b ~ & \textcolor{red}{ \text{ Distribute the }-1.} \end{aligned}\nonumber \]

We chose to use the “quicker” technique of “distributing” the −1. Here is our thinking:

1. Multiply −1 times a, getting −a.
2. Multiply −1 times +b, getting −b.

Exercise

Remove parentheses: −(4a − 3c)

Answer

−4a + 3c

The results in Example 10 and Example 11 show us how to negate a sum: Simply negate each term of the sum. Positive terms change to negative, negative terms turn to positive.

Negating a Sum

To negate a sum, simply negate each term of the sum. For example, if a and b are integers, then

−(a + b) = −a − b and − (a − b) = −a + b.

Example 12

Remove parentheses: −(5 − 7u + 3t).

Solution

Simply negate each term in the parentheses.

\[−(5 − 7u + 3t) = −5+7u − 3t\nonumber \]

Exercise

Remove parentheses: −(5 − 2x + 4y − 5z)

Answer

−5+2x − 4y + 5z

Exercises

In Exercises 1-20, use the associative and commutative properties of multiplication to simplify the expression.

1. 10(−4x)

2. 7(−8x)

3. (−10x)(−3)

4. (−5x)(−8)

5. −5(3x)

6. 9(6x)

7. (−4x)10

8. (−10x)(−6)

9. (5x)3

10. (3x)3

11. (5x)10

12. (−2x)(−10)

13. −9(−7x)

14. −10(5x)

15. 6(2x)

16. 3(−10x)

17. −8(−9x)

18. 3(−3x)

19. (6x)7

20. (−8x)(−5)

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In Exercises 21-44, simplify the expression.

21. 8(7x + 8)

22. −2(5x + 5)

23. 9(−2 + 10x)

24. −9(4 + 9x)

25. −(−2x + 10y − 6)

26. −(−6y + 9x − 7)

27. 2(10 + x)

28. 2(10 − 6x)

29. 3(3 + 4x)

30. 3(4 + 6x)

31. −(−5 − 7x + 2y)

32. −(4x − 8 − 7y)

33. 4(−6x + 7)

34. 6(4x + 9)

35. 4(8x − 9)

36. 10(−10x + 1)

37. −(4 − 2x − 10y)

38. −(−4x + 6 − 8y)

39. −(−5x +1+9y)

40. −(−10 − 5x − 4y)

41. −(6x + 2 − 10y)

42. −(6x + 4 − 10y)

43. −(−3y − 4+4x)

44. −(−7 − 10x + 7y)

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Answers

1. −40x

3. 30x

5. −15x

7. −40x

9. 15x

11. 50x

13. 63x

15. 12x

17. 72x

19. 42x

21. 56x + 64

23. −18 + 90x

25. 2x − 10y + 6

27. 20 + 2x

29. 9 + 12x

31. 5+7x − 2y

33. −24x + 28

35. 32x − 36

37. −4+2x + 10y

39. 5x − 1 − 9y

41. −6x − 2 + 10y

43. 3y + 4 − 4x