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CHAPTER 10: RATIONAL EXPRESSIONS Chapter Objectives By the end of this chapter, students should be able to: Evaluate rational expressions Obtain the excluded values of the expression Reduce rational expressions Multiply rational expressions with and without factoring Divide rational expressions with and without factoring Find least common denominators Add and subtract rational expressions with and without common denominators

Contents CHAPTER 10: RATIONAL EXPRESSIONS .................................................................................................... 319

SECTION 10.1: REDUCE RATIONAL EXPRESSIONS ................................................................................... 320

A. EVALUATE RATIONAL EXPRESSIONS ........................................................................................ 320

B. FIND EXCLUDED VALUES OF RATIONAL EXPRESSIONS ........................................................... 321

C. REDUCE RATIONAL EXPRESSIONS WITH MONOMIALS ........................................................... 322

D. REDUCE RATIONAL EXPRESSIONS WITH POLYNOMIALS ........................................................ 323

EXERCISES ......................................................................................................................................... 324

SECTION 10.2: MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS ....................................................... 325

A. MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS WITH MONOMIALS ................................... 325

B. MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS WITH POLYNOMIALS ................................ 326

C. MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS IN GENERAL ............................................... 327

EXERCISES ......................................................................................................................................... 328

SECTION 10.3 OBTAIN THE LOWEST COMMON DENOMINATOR ....................................................... 329

A. OBTAIN THE LCM IN ARITHMETIC REVIEW .............................................................................. 329

B. OBTAIN THE LCM WITH MONOMIALS ..................................................................................... 330

C. OBTAIN THE LCM WITH POLYNOMIALS ................................................................................... 330

D. REWRITE FRACTIONS WITH THE LOWEST COMMON .............................................................. 331

EXERCISES ......................................................................................................................................... 332

SECTION 10.4: ADD AND SUBTRACT RATIONAL EXPRESSIONS .......................................................... 333

A. ADD OR SUBTRACT RATIONAL EXPRESSIONS WITH A COMMON DENOMINATOR ............... 333

B. ADD AND SUBTRACT RATIONAL EXPRESSIONS WITH UNLIKE DENOMINATORS ................... 334

EXERCISE ........................................................................................................................................... 336

CHAPTER REVIEW ................................................................................................................................. 337

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SECTION 10.1: REDUCE RATIONAL EXPRESSIONS A. EVALUATE RATIONAL EXPRESSIONS

Definition

A rational expression is a ratio of two polynomials, i.e., a fraction where the numerator and denominator are polynomials.

MEDIA LESSON Evaluate rational expressions (Duration 4:18 )

View the video lesson, take notes and complete the problems below. Rational Expression: Quotient of two ______________________________________________________

a) −𝑥𝑥2−2𝑥𝑥−8𝑥𝑥−4

when 𝑥𝑥 = −4 b) 𝑥𝑥2−𝑥𝑥−6𝑥𝑥2+𝑥𝑥−12

when 𝑥𝑥 = 2

YOU TRY

Evaluate.

a) 𝑥𝑥2−4

𝑥𝑥2+6𝑥𝑥+8 𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑥𝑥 = −6

b) 3𝑥𝑥

𝑥𝑥2+12𝑥𝑥−2 when 𝑥𝑥 = −2

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B. FIND EXCLUDED VALUES OF RATIONAL EXPRESSIONS Rational expressions are special types of fractions, but still hold the same arithmetic properties. One property of fractions we recall is that the fraction is undefined when the denominator is zero.

Determine the excluded value(s) of a rational expression

Note: A rational expression is undefined when the denominator is zero.

Step 1. Set the denominator of the rational expression equal to zero.

Step 2. Solve the equation for the given variable.

Step 3. The values found in the previous step are the values excluded from the expression.

MEDIA LESSON Find excluded value(s) of a rational expression (Duration 2:24 )

View the video lesson, take notes and complete the problems below.

a) 𝑥𝑥2−13𝑥𝑥2+5𝑥𝑥

YOU TRY

Find the excluded value(s) of the expression.

a) −3𝑧𝑧𝑧𝑧+5

b) 𝑥𝑥2−13𝑥𝑥2+5𝑥𝑥

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C. REDUCE RATIONAL EXPRESSIONS WITH MONOMIALS Rational expressions are reduced, just as in arithmetic, even without knowing the value of the variable. When we reduce, we divide out common factors as we discussed with polynomial division with monomials. Now, we use factoring techniques and exponent properties to reduce rational expressions.

Reducing rational expressions

If 𝑃𝑃,𝑄𝑄,𝐾𝐾 are non-zero polynomials and 𝑃𝑃𝑃𝑃𝑄𝑄𝑃𝑃

is a rational expression, then

𝑷𝑷 . 𝑲𝑲𝑸𝑸 . 𝑲𝑲

= 𝑷𝑷𝑸𝑸

We call a rational expression irreducible if there are no more common factors among the numerator and denominator.

MEDIA LESSON Reduce monomials (Duration 2:44)

View the video lesson, take notes and complete the problems below.

Quotient rule of exponents: 𝑎𝑎𝑚𝑚

𝑎𝑎𝑛𝑛 =_______________

a) 16𝑎𝑎5

12𝑥𝑥9

b) 15𝑎𝑎3𝑏𝑏2

25𝑎𝑎𝑏𝑏5

It is important to note that we were only able to use the quotient rule when_______________________

____________________________________________________________________________________.

YOU TRY

Simplify.

a) 2𝑥𝑥2

4𝑥𝑥3

b) 15𝑥𝑥4𝑦𝑦2

25𝑥𝑥2𝑦𝑦6

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D. REDUCE RATIONAL EXPRESSIONS WITH POLYNOMIALS However, if there is a sum or difference in either the numerator or denominator, we first factor the numerator and denominator to obtain a product of factors, and then reduce.

MEDIA LESSON Reduce polynomials (Duration 5:00)

View the video lesson, take notes and complete the problems below. To reduce polynomials, we _______________________________ common _______________________.

This means we must first ________________________.

a) 2𝑥𝑥2+5𝑥𝑥−32𝑥𝑥2−5𝑥𝑥+2

Note: you can use the “bottoms-up” method to factor the binomials.

b) 9𝑥𝑥2−30𝑥𝑥+25

9𝑥𝑥2−25

YOU TRY

Simplify.

a) 𝟐𝟐𝟐𝟐𝟐𝟐𝟖𝟖𝟐𝟐−𝟏𝟏𝟏𝟏

b) 𝟗𝟗𝟖𝟖−𝟑𝟑𝟏𝟏𝟐𝟐𝟖𝟖−𝟏𝟏

c) 𝟖𝟖𝟐𝟐−𝟐𝟐𝟐𝟐𝟖𝟖𝟐𝟐+𝟐𝟐𝟖𝟖+𝟏𝟏𝟐𝟐

Warning: You cannot reduce terms, only factors. This means we cannot reduce anything with a “+” or “–” between the parts. In examples above, we are not allowed to divide out the 𝑥𝑥’s because they are terms (separated by + 𝑜𝑜𝑜𝑜 −) not factors (separated by multiplication).

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EXERCISES Evaluate the expression for the given value.

1) 4𝑣𝑣 + 26

𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑣𝑣 = 6

2) 𝑥𝑥−3

𝑥𝑥2−4𝑥𝑥+3 𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑥𝑥 = −4

3)

𝑏𝑏+2𝑏𝑏2+4𝑏𝑏+4

𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑏𝑏 = 0

4) 𝑏𝑏−33𝑏𝑏+9

𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑏𝑏 = −2 5) 𝑎𝑎+2

𝑎𝑎2+3𝑎𝑎+2 𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑎𝑎 = −1 6)

𝑛𝑛2−𝑛𝑛−6𝑛𝑛−3

𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑒𝑒 = 4

Find the excluded value(s).

7) 3𝑘𝑘2+30𝑘𝑘𝑘𝑘+10

8) 15𝑛𝑛2

10𝑛𝑛+25 9)

10𝑚𝑚2+8𝑚𝑚10𝑚𝑚

10) 𝑟𝑟2+3𝑟𝑟+25𝑟𝑟+10

11) 𝑏𝑏2+12𝑏𝑏+32 𝑏𝑏2+4𝑏𝑏−32

12) 27𝑝𝑝

18𝑝𝑝2−36𝑝𝑝

13) 𝑥𝑥+10

8𝑥𝑥2+80𝑥𝑥

14)

10𝑥𝑥+166𝑥𝑥+20

15) 6𝑛𝑛2−21𝑛𝑛6𝑛𝑛2+3𝑛𝑛

Simplify each expression.

16) 21𝑥𝑥2

18𝑥𝑥 17)

24𝑎𝑎40𝑎𝑎2

18) 32𝑥𝑥3

8𝑥𝑥4

19) 18𝑚𝑚−24

60 20)

204𝑝𝑝+2

21) 𝑥𝑥+1

𝑥𝑥2+8𝑥𝑥+7

22) 32𝑥𝑥2

28𝑥𝑥2+28𝑥𝑥 23)

𝑛𝑛2+4𝑛𝑛−12𝑛𝑛2−7𝑛𝑛+10

24) 9𝑣𝑣+54

𝑣𝑣2−4𝑣𝑣−60

25) 12𝑥𝑥2−42𝑥𝑥30𝑥𝑥2−42𝑥𝑥

26) 6𝑎𝑎−1010𝑎𝑎+4

27) 2𝑛𝑛2+19𝑛𝑛− 10

9𝑛𝑛+90

28) 𝑛𝑛−99𝑛𝑛−81

29) 28𝑚𝑚+12

36 30)

49𝑟𝑟+5656𝑟𝑟

31) 𝑏𝑏2+14𝑏𝑏+48𝑏𝑏2+15𝑏𝑏+56

32) 30𝑥𝑥−9050𝑥𝑥+40

33) 𝑘𝑘2−12𝑘𝑘+32

𝑘𝑘2−64

34) 9𝑝𝑝+18

𝑝𝑝2+4𝑝𝑝+4 35)

3𝑥𝑥2−29𝑥𝑥+405𝑥𝑥2−30𝑥𝑥−80

36) 8𝑚𝑚+1620𝑚𝑚−12

37) 2𝑥𝑥2−10𝑥𝑥+83𝑥𝑥2−7𝑥𝑥+4

38) 7𝑛𝑛2−32𝑛𝑛+16

4𝑛𝑛−16 39)

𝑛𝑛2+2𝑛𝑛+16𝑛𝑛+6

40) 4𝑘𝑘3− 2𝑘𝑘2−2𝑘𝑘9𝑘𝑘3− 18𝑘𝑘2+ 9𝑘𝑘

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SECTION 10.2: MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS We use the same method for multiplying and dividing fractions to multiply and divide rational expressions.

A. MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS WITH MONOMIALS Recall. When we multiply two fractions, we divide out the common factors, e.g.

109⋅ 2125

= 5⋅23⋅3⋅ 7⋅35⋅5

= 1415

We multiply rational expressions using the same method.

MEDIA LESSON Multiply and divide monomials (Duration 4:49)

View the video lesson, take notes and complete the problems below. With monomials, we can use ____________________________________________________________.

𝑎𝑎𝑚𝑚 ⋅ 𝑎𝑎𝑛𝑛=____________________

𝑎𝑎𝑚𝑚

𝑎𝑎𝑛𝑛 = _______________________

a) 6𝑥𝑥2𝑦𝑦5

5𝑥𝑥3⋅ 10𝑥𝑥4

3𝑥𝑥2𝑦𝑦7 b)

4𝑎𝑎5𝑏𝑏9𝑎𝑎4

÷ 6𝑎𝑎𝑏𝑏4

12𝑏𝑏2

YOU TRY

a) Multiply: 25𝑥𝑥2

8𝑦𝑦8 ∙ 24𝑦𝑦

4

55𝑥𝑥7

b) Divide: 𝑎𝑎4𝑏𝑏2

𝑎𝑎 ÷ 𝑏𝑏

4

4

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B. MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS WITH POLYNOMIALS When we multiply or divide polynomials in rational expressions, we first factor using factoring techniques, then reduce out the common factors.

Warning: We are not allowed to reduce terms, only factors.

MEDIA LESSON Multiply and divide rational expressions with polynomials (Duration 5:00 )

View the video lesson, take notes and complete the problems below. To divide out factors, we must first ___________________________!

a) 𝑥𝑥2+3𝑥𝑥+24𝑥𝑥−12

⋅ 𝑥𝑥2−5𝑥𝑥+6𝑥𝑥2−4

b) 3𝑥𝑥2+5𝑥𝑥−2𝑥𝑥2+3𝑥𝑥+2

÷ 6𝑥𝑥2+𝑥𝑥−1𝑥𝑥2−3𝑥𝑥−4

YOU TRY

a) Multiply: 𝑥𝑥2− 9

𝑥𝑥2+ 𝑥𝑥−20 ∙ 𝑥𝑥

2−8𝑥𝑥+163𝑥𝑥+9

b) Divide: 𝑥𝑥2−𝑥𝑥−12𝑥𝑥2−2𝑥𝑥−8

÷ 5𝑥𝑥2+15𝑥𝑥

𝑥𝑥2+𝑥𝑥−2

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C. MULTIPLY AND DIVIDE RATIONAL EXPRESSIONS IN GENERAL We can combine multiplying and dividing rational expressions in one expression, but, remember; we reciprocate the fraction that directly precedes the division sign and then change the division to multiplication. Lastly, we can reduce the common factors.

Warning: We are not allowed to reduce terms, only factors.

MEDIA LESSON Multiply and divide rational expressions together (Duration 4:53)

View the video lesson, take notes and complete the problems below. To divide: ___________________________________________________.

Be sure to ____________________before _________________________.

a) 𝑥𝑥2+3𝑥𝑥−10𝑥𝑥2+6𝑥𝑥+5

⋅ 2𝑥𝑥2−𝑥𝑥−3

2𝑥𝑥2+𝑥𝑥−6÷ 8𝑥𝑥+20

6𝑥𝑥+15

YOU TRY

Simplify.

a) 𝑎𝑎2+7𝑎𝑎+10𝑎𝑎2+ 6𝑎𝑎+5

∙ 𝑎𝑎+1𝑎𝑎2+4𝑎𝑎+4

÷ 𝑎𝑎−1𝑎𝑎+2

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EXERCISES Simplify each expression. Watch for special products to help with factoring more quickly.

1) 8𝑥𝑥2

9 ∙ 9

2 2)

9𝑛𝑛2𝑛𝑛

∙ 75𝑛𝑛

3) 5𝑥𝑥2

4 ∙ 6

5 4)

7(𝑚𝑚−6)𝑚𝑚−6

∙ 5𝑚𝑚(7𝑚𝑚−5)7𝑚𝑚−5

5) 7𝑟𝑟

7𝑟𝑟(𝑟𝑟+10) ÷ 𝑟𝑟−6(𝑟𝑟−6)2 6)

25𝑛𝑛+255

∙ 430𝑛𝑛+30

7) 𝑥𝑥−1035𝑥𝑥+21

÷ 735𝑥𝑥+21

8) 𝑥𝑥2−6𝑥𝑥−7𝑥𝑥+5

∙ 𝑥𝑥+5𝑥𝑥−7

9) 8𝑘𝑘

24𝑘𝑘2−40𝑘𝑘 ÷ 1

15𝑘𝑘−25 10) (𝑒𝑒 − 8) ∙ 6

10𝑛𝑛−80

11) 4𝑚𝑚+36𝑚𝑚+9

∙ 𝑚𝑚−55𝑚𝑚2 12)

3𝑥𝑥−612𝑥𝑥−24

∙ (𝑥𝑥 + 3)

13) 𝑏𝑏+2

40𝑏𝑏2−24𝑏𝑏 ∙ (5𝑏𝑏 − 3) 14)

𝑛𝑛−76𝑛𝑛−12

∙ 12−6𝑛𝑛𝑛𝑛2−13𝑛𝑛+42

15) 27𝑎𝑎+369𝑎𝑎+63

÷ 6𝑎𝑎+82

16) 𝑥𝑥2−12𝑥𝑥+32𝑥𝑥2−6𝑥𝑥−16

∙ 7𝑥𝑥2+14𝑥𝑥

7𝑥𝑥2+21𝑥𝑥

17) (10𝑚𝑚2 + 100𝑚𝑚) ∙ 18𝑚𝑚3−36𝑚𝑚2

20𝑚𝑚2−40𝑚𝑚 18)

10𝑏𝑏2

30𝑏𝑏+20 ∙ 30𝑏𝑏+20

2𝑏𝑏2+10𝑏𝑏

19) 10𝑝𝑝5

÷ 810

20) 6𝑥𝑥 (𝑥𝑥+4)𝑥𝑥−3

∙ (𝑥𝑥−3)(𝑥𝑥−6)6𝑥𝑥 (𝑥𝑥−6)

21) 𝑣𝑣−14

∙ 4𝑣𝑣2−11𝑣𝑣+10

22) 𝑝𝑝−8

𝑝𝑝2−12𝑝𝑝+32 ÷ 1

𝑝𝑝−10

23) 2𝑟𝑟𝑟𝑟+6

÷ 2𝑟𝑟7𝑟𝑟+42

24) 𝑣𝑣2+10𝑣𝑣+93𝑣𝑣+4

÷ 𝑣𝑣−93𝑣𝑣+4

25) 𝑘𝑘−7

𝑘𝑘2−𝑘𝑘−12 ∙ 7𝑘𝑘

2−28𝑘𝑘8𝑘𝑘2−56𝑘𝑘

26) 𝑛𝑛−7

𝑛𝑛2−2𝑛𝑛−35 ÷ 9𝑛𝑛+54

10𝑛𝑛+50

27) 𝑛𝑛2+2𝑛𝑛+1𝑛𝑛2−1

∙ 25𝑛𝑛2−16

5𝑛𝑛+4 28)

𝑥𝑥2−12𝑥𝑥−4

∙ 𝑥𝑥2−4𝑥𝑥2−𝑥𝑥−2

÷ 𝑥𝑥2+𝑥𝑥−23𝑥𝑥−6

29) 𝑎𝑎3+33

𝑎𝑎2+3𝑎𝑎𝑏𝑏+2𝑏𝑏2⋅ 3𝑎𝑎−6𝑏𝑏3𝑎𝑎+9

÷ 𝑎𝑎2−4𝑏𝑏2

𝑎𝑎+2𝑏𝑏 30)

𝑥𝑥2+3𝑥𝑥−10𝑥𝑥2+6𝑥𝑥+5

⋅ 2𝑥𝑥2−𝑥𝑥−3

2𝑥𝑥2+𝑥𝑥−6÷ 8𝑥𝑥+20

6𝑥𝑥+15

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SECTION 10.3 OBTAIN THE LOWEST COMMON DENOMINATOR As with fractions in arithmetic, the least common denominator or LCD is the lowest common multiple (LCM) of the denominators. Since rational expressions are fractions with polynomials, we use the LCD to add and subtract rational expression with different denominators. In this section, we obtain LCDs of rational expressions. First, let’s take a look at the method in finding the LCM in arithmetic.

A. OBTAIN THE LCM IN ARITHMETIC REVIEW To find the LCM using the prime factorization:

1) Find the prime factorization for each number by using the factor tree 2) Write each number in the exponential form 3) Collect all prime factors that show up in all numbers with the highest exponent 4) Multiply all the prime factors that collected in step 3 to find the LCM

MEDIA LESSON Determining the Least Common Multiple Using Prime Factorization (Duration 4:41)

View the video lesson, take notes and complete the problems below. Determine the least common multiple (LCM).

a) 16 and 18 b) 72 and 54

YOU TRY

Find LCM.

a) Find LCM of 3, 6, and 15 using the prime factorization method.

b) Find LCM of 25, 315 and 150 using the prime factorization method.

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B. OBTAIN THE LCM WITH MONOMIALS

MEDIA LESSON Find the LCM with monomials (Duration 2:55)

View the video lesson, take notes and complete the problems below.

To find the LCM/LCD of monomials:

Use ___________ factors with ________________ exponents.

Find the LCM of the monomials below.

a) 5𝑥𝑥3𝑦𝑦2 and 4𝑥𝑥2𝑦𝑦5 b) 7𝑎𝑎𝑏𝑏2𝑐𝑐 and 3𝑎𝑎3𝑏𝑏

YOU TRY

Find LCM:

a) 4𝑥𝑥2𝑦𝑦5 and 6𝑥𝑥4𝑦𝑦3𝑧𝑧6

b) 12𝑎𝑎2𝑏𝑏5 and 18𝑎𝑎𝑏𝑏𝑐𝑐

C. OBTAIN THE LCM WITH POLYNOMIALS We use the same method, but now we factor using factoring techniques to obtain the LCM between polynomials. Recall, all factors are contained in the LCM.

MEDIA LESSON Find the LCM of polynomials (Duration 4:45)

View the video lesson, take notes and complete the problems below.

To find the LCM/LCD of polynomials:

Use ___________ factors with ____________ exponents.

This means we must first ________________.

Find the LCM of the following polynomials.

a) 𝑥𝑥2 + 3𝑥𝑥 − 18 and 𝑥𝑥2 + 4𝑥𝑥 − 21 b) 𝑥𝑥2 − 10𝑥𝑥 + 25 and 𝑥𝑥2 − 𝑥𝑥 − 20

Chapter 10

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YOU TRY

Find the LCM of the following polynomials.

a) 𝑥𝑥2 + 2𝑥𝑥 − 3 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥2 − 𝑥𝑥 − 12

b) 𝑥𝑥2 − 10𝑥𝑥 + 25 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥2 − 14𝑥𝑥 + 45

D. REWRITE FRACTIONS WITH THE LOWEST COMMON

MEDIA LESSON Identify LCD and build up to matching denominators (Duration 4:59 )

View the video lesson, take notes and complete the problems below. Example:

a) 5𝑎𝑎4𝑏𝑏3𝑐𝑐

and 3𝑐𝑐

6𝑎𝑎2𝑏𝑏 b) 5𝑥𝑥

𝑥𝑥2−5𝑥𝑥−6 and

𝑥𝑥−2𝑥𝑥2+4𝑥𝑥+3

YOU TRY

Find the LCD between the two fractions. Rewrite each fraction with the LCD.

a) 5𝑎𝑎4𝑏𝑏3𝑐𝑐

𝑎𝑎𝑒𝑒𝑎𝑎 3𝑐𝑐6𝑎𝑎2𝑏𝑏

b) 5𝑥𝑥𝑥𝑥2−5𝑥𝑥−6

𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥−2𝑥𝑥2+4𝑥𝑥+3

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EXERCISES Find the equivalent numerator.

1) 38

= ?48

2) 𝑎𝑎𝑥𝑥

= ?𝑥𝑥𝑦𝑦

3) 𝑎𝑎5

= ?5𝑎𝑎

4) 2

𝑥𝑥+4= ?

𝑥𝑥2−16

5) (𝑥𝑥−4)(𝑥𝑥+2) = ?

𝑥𝑥2+5𝑥𝑥+6 6)

23𝑎𝑎2𝑏𝑏2𝑐𝑐

= ?9𝑎𝑎5𝑏𝑏2𝑐𝑐4

Find the lowest common multiple.

7) 2𝑎𝑎3, 6𝑎𝑎4𝑏𝑏2 𝑎𝑎𝑒𝑒𝑎𝑎 4𝑎𝑎3𝑏𝑏5 8) 𝑥𝑥2 − 3𝑥𝑥, 𝑥𝑥 − 3 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥

9) 𝑥𝑥 + 2 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥 − 4 10) 𝑥𝑥2 − 25 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥 + 5

11) 𝑥𝑥2 + 3𝑥𝑥 + 2 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥2 + 5𝑥𝑥 + 6 12) 5𝑥𝑥2𝑦𝑦 𝑎𝑎𝑒𝑒𝑎𝑎 25𝑥𝑥3𝑦𝑦5𝑧𝑧

13) 4𝑥𝑥 − 8, 𝑥𝑥 − 2 𝑎𝑎𝑒𝑒𝑎𝑎 4 14) 𝑥𝑥, 𝑥𝑥 − 7 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥 + 1

15) 𝑥𝑥2 − 9 𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥2 − 6𝑥𝑥 + 9 16) 𝑥𝑥2 − 7𝑥𝑥 + 10, 𝑥𝑥2 − 2𝑥𝑥 − 15,

𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥2 + 𝑥𝑥 − 6

Find the LCD and rewrite each fraction with the LCD.

17) 3𝑎𝑎5𝑏𝑏2

and 210𝑎𝑎3𝑏𝑏

18) 𝑥𝑥+2𝑥𝑥−3

and 𝑥𝑥−3𝑥𝑥+2

19) 𝑥𝑥

𝑥𝑥2−16 and

3𝑥𝑥𝑥𝑥2−8𝑥𝑥+16

20) 𝑥𝑥+1𝑥𝑥2−36

and 2𝑥𝑥+3𝑥𝑥2+12𝑥𝑥+36

21) 4𝑥𝑥

𝑥𝑥2−𝑥𝑥−6 and 𝑥𝑥+2

𝑥𝑥−3 22)

3𝑥𝑥𝑥𝑥−4

and 2𝑥𝑥+2

23) 5

𝑥𝑥2−6𝑥𝑥 , 2𝑥𝑥

and −3𝑥𝑥−6

24) 5𝑥𝑥+1

𝑥𝑥2−3𝑥𝑥−10 and 4

𝑥𝑥−5

25) 3𝑥𝑥+1

𝑥𝑥2−𝑥𝑥−12 and 2𝑥𝑥

𝑥𝑥2+4𝑥𝑥+3 26)

3𝑥𝑥𝑥𝑥2−6𝑥𝑥+8

𝑥𝑥−2𝑥𝑥2+𝑥𝑥−20

and 5𝑥𝑥2+3𝑥𝑥−10

Chapter 10

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SECTION 10.4: ADD AND SUBTRACT RATIONAL EXPRESSIONS Adding and subtracting rational expressions are identical to adding and subtracting with fractions. Recall, when adding with a common denominator, we add across numerators and keep the same denominator. This is the same method we use with rational expressions. Note, methods never change, only problems.

Helpful tips when adding and subtracting rational expressions:

For adding and subtracting with rational expressions, here are some helpful tips: • Identify the denominators: are they the same or different? • Combine the rational expressions into one expression. • Once combined into one expression, then reduce the fraction, if possible. • A fraction is reducible only if there is a GCF in the numerator.

A. ADD OR SUBTRACT RATIONAL EXPRESSIONS WITH A COMMON DENOMINATOR Recall. We can use the same properties for adding or subtracting fractions with common denominators also for adding and subtracting rational expressions with common denominators:

𝑎𝑎𝑐𝑐

±𝑏𝑏𝑐𝑐

=𝑎𝑎 ± 𝑏𝑏𝑐𝑐

MEDIA LESSON Add/ subtract rational expressions with common denominator (Duration 5:00)

View the video lesson, take notes and complete the problems below. Add/subtract rational expressions

• Add the ___________________________ and keep the __________________________

• When subtracting, we will first _____________________ the negative.

• Don’t forget to ___________________

Example:

a) 𝑥𝑥2+4𝑥𝑥𝑥𝑥2−2𝑥𝑥−15

+ 𝑥𝑥+6𝑥𝑥2−2𝑥𝑥−15

b) 𝑥𝑥2+2𝑥𝑥2𝑥𝑥2−9𝑥𝑥−5

− 6𝑥𝑥+52𝑥𝑥2−9𝑥𝑥−5

Chapter 10

Jan ‘19 334

YOU TRY

Evaluate.

a) Add: 𝑥𝑥−4

𝑥𝑥2−2𝑥𝑥−8+ 𝑥𝑥+8

𝑥𝑥2−2𝑥𝑥−8

b) Subtract: 6𝑥𝑥−123𝑥𝑥−6

− 15𝑥𝑥−63𝑥𝑥−6

B. ADD AND SUBTRACT RATIONAL EXPRESSIONS WITH UNLIKE DENOMINATORS Recall. We can use the same properties for adding and subtracting integer fractions with unlike denominators for adding and subtracting rational expressions with unlike denominators:

𝑎𝑎𝑏𝑏

±𝑐𝑐𝑎𝑎

=𝑎𝑎𝑎𝑎 ± 𝑏𝑏𝑐𝑐𝑏𝑏𝑎𝑎

MEDIA LESSON Add rational expressions with different denominators (Duration 4:56)

View the video lesson, take notes and complete the problems below. Add/subtract rational expressions with different denominators

To add or subtract, we __________________the denominators by ___________ by the missing _______

This means we have to __________________ to find the LCD.

Example:

a) 2𝑥𝑥

𝑥𝑥2−9+ 5

𝑥𝑥2+𝑥𝑥−6

Chapter 10

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MEDIA LESSON Subtract rational expressions with different denominators (Duration 5:00)

View the video lesson, take notes and complete the problems below. Example:

b) 2𝑥𝑥+7

𝑥𝑥2−2𝑥𝑥−3− 3𝑥𝑥−2

𝑥𝑥2+6𝑥𝑥+5

Warning: We are not allowed to reduce terms, only factors.

YOU TRY

a) Add 7𝑎𝑎3𝑎𝑎2𝑏𝑏

+ 4𝑏𝑏6𝑎𝑎𝑏𝑏4

.

b) Subtract 45𝑎𝑎− 7𝑏𝑏

4𝑎𝑎2 .

c) Add 6

8𝑎𝑎+4+ 3𝑎𝑎

8 .

d) Subtract 𝑥𝑥+1𝑥𝑥−4

− 𝑥𝑥+1𝑥𝑥2−7𝑥𝑥+12

.

Chapter 10

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EXERCISE Add or subtract the rational expressions. Simplify completely.

1) 2

𝑎𝑎+3+ 4

𝑎𝑎+3 2)

𝑡𝑡2+4𝑡𝑡𝑡𝑡−1

+ 2𝑡𝑡−7𝑡𝑡−1

3) 56𝑟𝑟− 5

8𝑟𝑟 4)

89𝑡𝑡2

+ 56𝑡𝑡2

5) 𝑎𝑎+22− 𝑎𝑎−4

4 6)

𝑥𝑥−14𝑥𝑥

− 2𝑥𝑥+3𝑥𝑥

7) 5𝑥𝑥+3𝑦𝑦2𝑥𝑥2𝑦𝑦

− 3𝑥𝑥+4𝑦𝑦𝑥𝑥𝑦𝑦2

8) 2𝑧𝑧𝑧𝑧−1

− 3𝑧𝑧𝑧𝑧+1

9) 8

𝑥𝑥2−4− 3

𝑥𝑥+2 10)

𝑡𝑡𝑡𝑡−3

− 54𝑡𝑡−12

11) 2

5𝑥𝑥2+5𝑥𝑥− 4

3𝑥𝑥+3 12)

𝑡𝑡𝑦𝑦−𝑡𝑡

− 𝑦𝑦𝑦𝑦+𝑡𝑡

13) 𝑥𝑥

𝑥𝑥2+5𝑥𝑥+6− 2

𝑥𝑥2+3𝑥𝑥+2 14) 2𝑥𝑥

𝑥𝑥2−1− 4

𝑥𝑥2+2𝑥𝑥−3

15) 4−𝑎𝑎2

𝑎𝑎2−9− 𝑎𝑎−2

3−𝑎𝑎 16)

𝑥𝑥2

𝑥𝑥−2− 6𝑥𝑥−8

𝑥𝑥−2

17) 7𝑥𝑥𝑦𝑦2

+ 3𝑥𝑥2𝑦𝑦

18) 2𝑎𝑎−13𝑎𝑎2

+ 5𝑎𝑎+19𝑎𝑎

19) 2𝑐𝑐−𝑑𝑑𝑐𝑐2𝑑𝑑

− 𝑐𝑐+𝑑𝑑𝑐𝑐𝑑𝑑2

20) 2

𝑥𝑥−1+ 2

𝑥𝑥+1

21) 2

𝑥𝑥−5+ 3

4𝑥𝑥 22)

4𝑥𝑥𝑥𝑥2−25

+ 𝑥𝑥𝑥𝑥+5

23) 3𝑎𝑎

4𝑎𝑎−20+ 9𝑎𝑎

6𝑎𝑎−30 24)

2𝑥𝑥𝑥𝑥2−1

− 3𝑥𝑥2+5𝑥𝑥+4

25) 2𝑥𝑥

𝑥𝑥2−9+ 5

𝑥𝑥2+𝑥𝑥−6 26)

4𝑥𝑥𝑥𝑥2−2𝑥𝑥−3

− 3𝑥𝑥2−5𝑥𝑥+6

27) 𝑥𝑥−1

𝑥𝑥2+3𝑥𝑥+2+ 𝑥𝑥+5

𝑥𝑥2+5𝑥𝑥+4 28)

3𝑥𝑥+23𝑥𝑥+6

+ 𝑥𝑥4−𝑥𝑥2

29) 2𝑟𝑟

𝑟𝑟2−𝑠𝑠2+ 1

𝑟𝑟+𝑠𝑠− 1

𝑟𝑟−𝑠𝑠 30)

𝑥𝑥+2𝑥𝑥2−4𝑥𝑥+3

+ 4𝑥𝑥+5𝑥𝑥2+4𝑥𝑥−5

Chapter 10

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CHAPTER REVIEW KEY TERMS AND CONCEPTS

Look for the following terms and concepts as you work through the workbook. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the text or in the media lesson.

Rational expression

Undefined rational expression

Evaluate the expression for the given value. Demonstrate your understanding.

1) 2𝑥𝑥+4𝑥𝑥

when 𝑥𝑥 = 2

2) 𝑥𝑥2+𝑥𝑥+1𝑥𝑥2+1

when 𝑥𝑥 = −2 3) −𝑥𝑥3+2

4 when 𝑥𝑥 = −1

Find the excluded value(s). Demonstrate your understanding.

4) 1+𝑥𝑥𝑥𝑥

5) 𝑥𝑥2−4𝑥𝑥+2

6) 3𝑥𝑥+9𝑥𝑥−3

Simplify each expression. Demonstrate your understanding.

7) 8𝑥𝑥8𝑦𝑦5

12𝑥𝑥𝑦𝑦4

8) 4𝑥𝑥+12

12𝑥𝑥+24𝑥𝑥2 9)

𝑥𝑥2+10𝑥𝑥+9𝑥𝑥2+17𝑥𝑥+72

10) 35𝑥𝑥+3521𝑥𝑥+7

11) 8𝑥𝑥3𝑥𝑥

÷ 47

12) 9𝑚𝑚5𝑚𝑚2 ∙

72

13) 6𝑥𝑥(𝑥𝑥+4)𝑥𝑥−3

∙ (𝑥𝑥−3)(𝑥𝑥−6)6𝑥𝑥(𝑥𝑥−6)

14) 2𝑛𝑛2−12𝑛𝑛−54

𝑛𝑛+7 ÷ (2𝑒𝑒 + 6) 15)

𝑥𝑥2−7𝑥𝑥+10𝑥𝑥−2

∙ 𝑥𝑥+10𝑥𝑥2−𝑥𝑥−20

Find the equivalent numerator. Demonstrate your understanding.

16) 52𝑥𝑥2

= ?8𝑥𝑥3𝑦𝑦

17) 4

3𝑎𝑎5𝑏𝑏2𝑐𝑐4= ?

9𝑎𝑎5𝑏𝑏2𝑐𝑐4 18)

𝑥𝑥−6𝑥𝑥+3

= ?𝑥𝑥2−2𝑥𝑥−15

Find the lowest common multiple. Demonstrate your understanding.

19) 𝑥𝑥2 − 9, 𝑥𝑥 − 3,𝑎𝑎𝑒𝑒𝑎𝑎 𝑥𝑥2

20) 𝑥𝑥 + 3, 𝑥𝑥 − 3,𝑎𝑎𝑒𝑒𝑎𝑎 2 21) 10, 40,𝑎𝑎𝑒𝑒𝑎𝑎 5

Chapter 10

Jan ‘19 338

Find the LCD and rewrite each fraction with the LCD. Demonstrate your understanding.

22) 𝑥𝑥+3𝑥𝑥2−16

and 𝑥𝑥𝑥𝑥2+1

23) 4

5𝑥𝑥𝑦𝑦2 and 2

15𝑦𝑦 24)

2𝑥𝑥2+5𝑥𝑥+6

and 3

𝑥𝑥+2

Add or subtract the rational expressions. Simplify completely. Demonstrate your understanding.

25) 2𝑥𝑥2+3

𝑥𝑥2−6𝑥𝑥+5− 𝑥𝑥2−5𝑥𝑥+9

𝑥𝑥2−6𝑥𝑥+5

26) 𝑥𝑥

𝑥𝑥2+15𝑥𝑥+56− 7

𝑥𝑥2+13𝑥𝑥+42 27)

5𝑥𝑥𝑥𝑥2−𝑥𝑥−6

− 18𝑥𝑥2−9

28) 𝑥𝑥+1

𝑥𝑥2−2𝑥𝑥−35+ 𝑥𝑥+6

𝑥𝑥2+7𝑥𝑥+10

29) 2𝑧𝑧

1−2𝑧𝑧+ 3𝑧𝑧

2𝑧𝑧+1− 3

4𝑧𝑧2−1 30)

3𝑥𝑥−8𝑥𝑥2+6𝑥𝑥+8

+ 2𝑥𝑥−3𝑥𝑥2+3𝑥𝑥+2