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We begin our discussion with the definition of a *term*.

Definition: Term

A **term **is a single number or variable, or it can be the product of a number (called its coefficient) and one or more variables (called its variable part). The terms in an algebraic expression are separated by addition symbols.

Example 1

Identify the terms in the algebraic expression

\[ 3x^2 + 5xy + 9y^2 + 12\nonumber \]

For each term, identify its coefficient and variable part.

**Solution**

In tabular form, we list each term of the expression 3x^2 + 5xy + 9y^2 + 12, its coefficient, and its variable part.

Term | Coefficient | Variable Part |
---|---|---|

3x^{2} | 3 | x^{2} |

5xy | 5 | xy |

9y^{2} | 9 | y^{2} |

12 | 12 | None |

Example 2

Identify the terms in the algebraic expression

\[ a^3 − 3a^2b + 3ab^2 − b^3\nonumber \]

For each term, identify its coefficient and variable part.

**Solution**

The first step is to write each difference as a sum, because the terms of an expression are defined above to be those items separated by addition symbols.

\[ a^3 + (−3a^2b)+3ab^2 + (−b^3)\nonumber \]

In tabular form, we list each term of the expression \(a^3+(−3a^2b)+3ab^2+(−b^3)\), its coefficient, and its variable part.

Term | Coefficient | Variable Part |
---|---|---|

a^{3} | 1 | a^{3} |

-3a^{2}b | −3 | a^{2}b |

3ab^{2} | 3 | ab^{2} |

−b^{3} | −1 | b^{3} |

Exercise

How many terms are in the algebraic expression \(11 − a^2 − 2ab + 3b^2a\)?

**Answer**-
4

## Like Terms

We define what is meant by “like terms” and “unlike terms.”

Definition

**Like and Unlike Terms.** The variable parts of two terms determine whether the terms are like terms or unlike terms.

**Like Terms**. Two terms are called like terms if they have identical variable parts, which means that the terms must contain the same variables raised to the same exponential powers.

**Unlike Terms. **Two terms are called unlike terms if their variable parts are different.

Example 3

Classify each of the following pairs as either *like* *terms* or *unlike* *terms*: (a) 3*x* and −7*x*, (b) 2*y* and 3*y*^{2}, (c) −3*t* and 5*u*, and (d) −4*a*^{3} and 3*a*^{3}.

**Solution**

Like terms must have *identical* variable parts.

- 3
*x*and −7*x*have identical variable parts. They are “like terms.” - 2
*y*and 3*y*^{2}do not have identical variable parts (the exponents differ). They are “unlike terms.” - −3
*t*and 5*u*do not have identical variable parts (different variables). They are “unlike terms.” - −4
*a*^{3}and 3*a*^{3}have identical variable parts. They are “like terms.”

Exercise

Are −3*xy* and 11*xy* *like* or *unlike* terms?

**Answer**-
*Like*terms

## Combining Like Terms

When using the distributive property, it makes no difference whether the multiplication is on the left or the right, one still distributes the multiplication times each term in the parentheses.

Distributive Property

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

*a*(*b* + *c*) = *ab* + *ac* and (*b* + *c*)*a* = *ba* + *ca*.

In either case, you distribute a times each term of the sum.

“Like terms” can be combined and simplified. The tool used for combining like terms is the distributive property. For example, consider the expression 3*y* + 7*y*, composed of two “like terms” with a common variable part. We can use the distributive property and write

\[3y+7y=(3+7)y\nonumber \]

Note that we are using the distributive property in reverse, “factoring out” the common variable part of each term. Checking our work, note that if we redistribute the variable part y times each term in the parentheses, we are returned to the original expression 3*y *+ 7*y*.

Example 4

Use the distributive property to combine like terms (if possible) in each of the following expressions: (a) −5*x*^{2} − 9*x*^{2}, (b) −5*ab* + 7*ab*, (c) 4*y*^{3} − 7*y*^{2}, and (d) 3*xy*^{2} − 7*xy*^{2}.

**Solution**

If the terms are “like terms,” you can use the distributive property to “factor out” the common variable part.

a) Factor out the common variable part *x*^{2}.

\[ \begin{aligned} -5x^2 -9x^2 =(-5-9)x^2 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ = -14x^2 ~ & \textcolor{red}{ \text{ Simplify: } -5-9=-5+(-9) = -14.} \end{aligned}\nonumber \]

b) Factor out the common variable part *ab*.

\[ \begin{aligned} -5ab +7ab = (-5+7)ab ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =2ab ~ & \textcolor{red}{ \text{ Simplify: } -5+7 = 2.} \end{aligned}\nonumber \]

c) The terms in the expression 4*y*^{3} − 7*y*^{2} have different variable parts (the exponents are different). These are “unlike terms” and cannot be combined.

d) Factor out the common variable part *xy*^{2}.

\[ \begin{aligned} 3xy^2 - 7xy^2 =(3-7)xy^2 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =-4xy^2 ~ & \textcolor{red}{ \text{ Simplify: } 3-7=3+(-7)=-4.} \end{aligned}\nonumber \]

Exercise

Simplify: −8*z* − 11*z*

**Answer**-
−19

*z*

## Speeding Things Up a Bit

Once you’ve written out all the steps for combining like terms, like those shown in Example 4, you can speed things up a bit by following this rule:

Combining Like Terms

To combine like terms, simply add their coefficients and keep the common variable part.

Thus for example, when presented with the sum of two like terms, such as in 5*x*+ 8*x*, simply add the coefficients and repeat the common variable part; that is, 5*x* + 8*x* = 13*x*.

Example 5

Combine like terms:

- −9
*y*− 8y, - −3
*y*^{5}+ 4*y*^{5}and - −3
*u*^{2}+ 2*u*^{2}.

**Solution**

a) Add the coefficients and repeat the common variable part. Therefore,

\[−9y − 8y = −17y.\nonumber \]

b) Add the coefficients and repeat the common variable part. Therefore,

\[−3y^5 + 4y^5 = 1y^5.\nonumber \]

However, note that 1*y*^{5} = *y*^{5}. Following the rule that the final answer should use as few symbols as possible, a better answer is −3*y*^{5} + 4*y*^{5} = *y*^{5}.

c) Add the coefficients and repeat the common variable part. Therefore,

\[−3u^2 + 2u^2 = (−1)u^2.\nonumber \]

However, note that (−1)*u*^{2} = −*u*^{2}. Following the rule that the final answer should use as few symbols as possible, a better answer is −3*u*^{2}+ 2*u*^{2} = −*u*^{2}.

Exercise

Combine: −3*x*, and − 4*x*^{2}

**Answer**-
−7

*x*^{2}

## Simplify

A frequently occurring instruction asks the reader to *simplify* an expression.

Simplify

The instruction simplify is a generic term that means “try to write the expression in its most compact form, using the fewest symbols possible.”

One way you can accomplish this goal is by combining like terms when they are present.

Example 6

Simplify: 2*x* + 3*y* − 5*x* + 8*y*.

**Solution**

Use the commutative property to reorder terms and the associative and distributive properties to regroup and combine like terms.

\[ \begin{aligned} 2x + 3y - 5x + 8y = (2x - 5x) + (3y + 8y) ~ & \textcolor{red}{ \text{ Reorder and regroup.}} \\ = -3x + 11y ~ & \textcolor{red}{ \text{ Combine like terms:}} \\ ~ & \textcolor{red}{ 2x - 5x = -3x \text{ and } 3y + 8y = 11y.} \end{aligned}\nonumber \]

**Alternate solution**

Of course, you do not need to show the regrouping step. If you are more comfortable combining like terms in your head, you are free to present your work as follows:

\[2x + 3y − 5x + 8y = −3x + 11y.\nonumber \]

Exercise

Simplify: −3*a* + 4*b* − 7*a* − 9*b*

**Answer**-
−10

*a*− 5*b*

Example 7

Simplify: −2*x* − 3 − (3*x* + 4).

**Solution**

First, distribute the negative sign.

\[ \begin{aligned} -2x-3-(3x+4)= -2x-3-3x-4 ~ & \textcolor{red}{-(3x+4)=-3x-4.} \end{aligned}\nonumber \]

Next, use the commutative property to reorder, then the associative property to regroup. Then combine like terms.

\[ \begin{aligned} =(-2x-3x)+(-3-4) ~ & \textcolor{red}{ \text{ Reorder and regroup.}} \\ =-5x+(-7) ~ & \textcolor{red}{ \text{ Combine like terms:}} \\ ~ & \textcolor{red}{ -2x-3x=-5x.} \\ =-5x-7 ~ & \textcolor{red}{ \text{ Simplify:}} \\ ~ & \textcolor{red}{-5x+(-7)=-5x-7.} \end{aligned}\nonumber \]

**Alternate solution**

You may skip the second step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:

\[ \begin{aligned} -2x-3-(3x+4) = -2x-3-3x-4 ~ & \textcolor{red}{ \text{ Distribute negative sign.}} \\ =-5x-7 ~ & \textcolor{red}{ \text{ Combine like terms.}} \end{aligned}\nonumber \]

Exercise

Simplify: −9*a* − 4 − (4*a* − 8)

**Answer**-
−13

*a*+ 4

Example 8

Simplify: 2(5 − 3*x*) − 4(*x* + 3).

**Solution**

Use the distributive property to expand, then use the commutative and associative properties to group the like terms and combine them.

\[ \begin{aligned} 2(5-3x)-4(x+3) = 10-6x-4x-12 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =(-6x-4x)+(10-12) ~ & \textcolor{red}{ \text{ Group like terms.}} \\ =-10x-2 ~ & \textcolor{red}{ \text{ Combine like terms: }} \\ ~ & \textcolor{red}{-6x-4x=-10x \text{ and} \\ ~ & \textcolor{red}{10-12=-2.} \end{aligned}\nonumber \]

**Alternate solution **

You may skip the second step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:

\[ \begin{aligned} 2(5-3x)-4(x+3) = 10-6x-4x-12 ~ & \textcolor{red}{ \text{ Distribute.}} \\ =-10x-2 ~& \textcolor{red}{ \text{ Combine like terms.}} \end{aligned}\nonumber \]

Exercise

Simplify: −2(3*a* − 4) − 2(5 − *a*)

**Answer**-
−4

*a*− 2

Example 9

Simplify: −8(3*x*^{2}*y* − 9*xy*) − 8(−7*x*^{2} − 8*xy*)

**Solution**

We will proceed a bit quicker with this solution, using the distributive property to expand, then combining like terms mentally.

\[ \begin{aligned} -8(3x^2y-9xy) -8(-7x^2y-8xy)=-24x^2y+72xy+56x^2y+64xy \\ = 32x^2y+136xy \end{aligned}\nonumber \]

Exercise

Simplify: (*a*^{2} − 2*ab*) − 2(3*ab* + *a*^{2})

**Answer**-
−

*a*^{2}− 8*ab*

## Applications

We can simplify a number of useful formulas by combining like terms.

Example 10

Find the perimeter P of the (a) rectangle and (b) square pictured below. Simplify your answer as much as possible.

**Solution**

The perimeter of any polygonal figure is the sum of the lengths of its sides.

a) To find the perimeter *P* of the rectangle, sum its four sides.

\[P = L + W + L + W.\nonumber \]

Combine like terms.

\[P = 2L + 2W.\nonumber \]

b) To find the perimeter *P* of the square, sum its four sides.

\[P = s + s + s + s.\nonumber \]

Combine like terms.

\[P = 4s.\nonumber \]

Exercise

A regular hexagon has six equal sides, each with length *x*. Find its perimeter in terms of *x*.

**Answer**-
*P*= 6*x*

Sometimes it is useful to replace a variable with an expression containing another variable.

Example 11

The length of a rectangle is three feet longer than twice its width. Find the perimeter *P* of the rectangle in terms of its width alone.

**Solution**

From the previous problem, the perimeter of the rectangle is given by

\[P = 2L + 2W,\nonumber \]

where *L* and *W* are the length and width of the rectangle, respectively. This equation gives the perimeter in terms of its length and width, but we’re asked to get the perimeter in terms of the width alone. However, we’re also given the fact that the length is three feet longer than twice the width.

\[ \begin{array} \colorbox{cyan}{Length} & \text{ is } & \colorbox{cyan}{Three Feet} & \text{ longer than } & \colorbox{cyan}{Twice the Width} \\ L & = & 3 & + & 2W \end{array}\nonumber \]

Because *L* = 3+2*W*, we can replace *L* with 3+2*W* in the perimeter equation 3.1.

\[P = 2L + 2W\nonumber \]

\[P = 2(3 + 2W)+2W\nonumber \]

Use the distributive property, then combine like terms.

\[P =6+4W + 2W\nonumber \]

\[P =6+6W.\nonumber \]

This last equation gives the perimeter *P* in terms of the width *W* alone.

Exercise

The length *L* of a rectangle is 5 meters longer than twice its width *W*. Find the perimeter *P* of the rectangle in terms of its width *W*.

**Answer**-
Answer:

*P*= 6*W*+ 10

Example 12

The width of a rectangle is two feet less than its length. Find the perimeter *P* of the rectangle in terms of its length alone.

**Solution**

Again, the perimeter of a rectangle is given by the equation

\[P = 2L + 2W,\nonumber \]

where *L* and *W* are the length and width of the rectangle, respectively. This equation gives the perimeter in terms of its length and width, but we’re asked to get the perimeter in terms of the length alone.

However, we’re also given the fact that the width is two feet less than the length.

\[ \begin{aligned} \colorbox{cyan}{Width} & \text{ is } & \colorbox{cyan}{Length} & \text{ minus } & \colorbox{cyan}{Two feet} \\ W & = & L & - & 2 \end{aligned}\nonumber \]

Because *W* = *L*−2, we can replace *W* with *L*−2 in the perimeter equation 3.2.

\[P = 2L + 2W\nonumber \]

\[P = 2L + 2(L − 2)\nonumber \]

Use the distributive property, then combine like terms.

\[P = 2L + 2L − 4\nonumber \]

\[P = 4L − 4.\nonumber \]

This last equation gives the perimeter *P* in terms of the length *L* alone.

Exercise

The width *W* of a rectangle is 5 feet less than twice its width *L*. Find the perimeter *P* of the rectangle in terms of its length *L*.

**Answer**-
*P*= 6*L*− 10

## Exercises

In Exercises 1-16, combine like terms by first using the distributive property to factor out the common variable part, and then simplifying.

1. 17xy^{2} + 18xy^{2} + 20xy^{2}

2. 13xy − 3xy + xy

3. −8xy^{2} − 3xy^{2} − 10xy^{2}

4. −12xy − 2xy + 10xy

5. 4xy − 20xy

6. −7y^{3} + 15y^{3}

7. 12r − 12r

8. 16s − 5s

9. −11x − 13x + 8x

10. −9r − 10r + 3r

11. −5q + 7q

12. 17n + 15n

13. r − 13r − 7r

14. 19m + m + 15m

15. 3x^{3} − 18x^{3}

16. 13x^{2}y + 2x^{2}y

In Exercises 17-32, combine like terms by first rearranging the terms, then using the distributive property to factor out the common variable part, and then simplifying.

17. −8 + 17n + 10 + 8n

18. 11 + 16s − 14 − 6s

19. −2x^{3} − 19x^{2}y − 15x^{2}y + 11x^{3}

20. −9x^{2}y − 10y^{3} − 10y^{3} + 17x^{2}y

21. −14xy − 2x^{3} − 2x^{3} − 4xy

22. −4x^{3} + 12xy + 4xy − 12x^{3}

23. −13 + 16m + m + 16

24. 9 − 11x − 8x + 15

25. −14x^{2}y − 2xy^{2} + 8x^{2}y + 18xy^{2}

26. −19y^{2} + 18y^{3} − 5y^{2} − 17y^{3}

27. −14x^{3} + 16xy + 5x^{3} + 8xy

28. −16xy + 16y^{2} + 7xy + 17y^{2}^{ }

29. 9n + 10 + 7 + 15n

30. −12r + 5 + 17 + 17r

31. 3y +1+6y + 3

32. 19p +6+8p + 13

In Exercises 33-56, simplify the expression by first using the distributive property to expand the expression, and then rearranging and combining like terms mentally.

33. −4(9x^{2}y + 8) + 6(10x^{2}y − 6)

34. −4(−4xy + 5y^{3}) + 6(−5xy − 9y^{3})

35. 3(−4x^{2} + 10y^{2}) + 10(4y^{2} − x^{2})

36. −7(−7x^{3} + 6x^{2}) − 7(−10x^{2} − 7x^{3})

37. −s + 7 − (−1 − 3s)

38. 10y − 6 − (−10 − 10y)

39. −10q − 10 − (−3q + 5)

40. −2n + 10 − (7n − 1)

41. 7(8y + 7) − 6(8 − 7y)

42. −6(−5n − 4) − 9(3 + 4n)

43. 7(10x^{2} − 8xy^{2}) − 7(9xy^{2} + 9x^{2})

44. 10(8x^{2}y − 10xy^{2}) + 3(8xy^{2} + 2x^{2}y)

45. −2(6 + 4n) + 4(−n − 7)

46. −6(−2 − 6m) + 5(−9m + 7)

47. 8 − (4 + 8y)

48. −1 − (8 + s)

49. −8(−n + 4) − 10(−4n + 3)

50. 3(8r − 7) − 3(2r − 2)

51. −5 − (10p + 5)

52. −1 − (2p − 8)

53. 7(1 + 7r) + 2(4 − 5r)

54. (5 − s) + 10(9 + 5s)

55. −2(−5 − 8x^{2}) − 6(6)

56. 8(10y^{2} + 3x^{3}) − 5(−7y^{2} − 7x^{3})

57. The length L of a rectangle is 2 feet longer than 6 times its width W. Find the perimeter of the rectangle in terms of its width alone.

58. The length L of a rectangle is 7 feet longer than 6 times its width W. Find the perimeter of the rectangle in terms of its width alone.

59. The width W of a rectangle is 8 feet shorter than its length L. Find the perimeter of the rectangle in terms of its length alone.

60. The width W of a rectangle is 9 feet shorter than its length L. Find the perimeter of the rectangle in terms of its length alone.

61. The length L of a rectangle is 9 feet shorter than 4 times its width W. Find the perimeter of the rectangle in terms of its width alone.

62. The length L of a rectangle is 2 feet shorter than 6 times its width W. Find the perimeter of the rectangle in terms of its width alone.

## Answers

1. 55xy^{2}

3. −21xy^{2}

5. −16xy

7. 0

9. −16x

11. 2q

13. −19r

15. −15x^{3}

17. 2 + 25n

19. 9x^{3} − 34x^{2}y

21. −18xy − 4x^{3}

23. 3 + 17m

25. −6x^{2}y + 16xy^{2}

27. −9x^{3} + 24xy

29. 24n + 17

31. 9y + 4

33. 24x^{2}y − 68

35. −22x^{2} + 70y^{2}

37. 2s + 8

39. −7q − 15

41. 98y + 1

43. 7x^{2} − 119xy^{2}

45. −40 − 12n

47. 4 − 8y

49. 48n − 62

51. −10 − 10p

53. 15 + 39r

55. −26 + 16x^{2 }

57. 4 + 14W

59. 4L − 16

61. 10W − 18