Review Factoring Polynomials€¦ · by multiplying the numerator and denominator by the same...

Chapter 4 Math 3201 1

Review Factoring Polynomials:

1. GCF ex. 𝐴) 15𝑥2 − 5𝑥 𝐴) 24𝑥2 + 9𝑥

2. Difference of Squares 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

ex. A) 9𝑥2 − 16 B) 𝑥2 − 1 𝐶) 2𝑥2 − 98𝑦2

3. Trinomials ex. A) 𝑥2 + 5𝑥 − 24 B) 𝑥2 − 13𝑥 + 12 C) 3𝑥2 + 15𝑥 + 12

Solving Polynomials:

1. A) (2x – 5)(x – 1) = 0 B) 4𝑥2 − 6𝑥 = 0 C) 𝑥2 − 16 = 0


Practice Problems: Factor Completely

1. 9𝑥 + 18 2. 11𝑥2 − 6𝑥 3. 𝑤2 − 𝑢2

4. 925 2 x 5. 264 x 6. 9𝑚2𝑛2 − 64𝑝2

7. 12 x 8. 644 2 x 9 . bba 2

10. 1622 4 x 11. 𝑥2 − 10𝑥 + 25 12. 𝑥2 + 10𝑥 + 16

13. 𝑥2 − 4𝑥 − 12 14. 12132 xx 15. 3072 xx

16. 49142 xx 17. 2762 xx 18. 122 xx

19. 963 2 xx 20. 40364 2 xx 21. 21243 2 xx

B. Solve

1. 3𝑥 − 15 = 0 2. 16𝑥2 − 4𝑥 = 0 3. 4𝑥2 − 1 = 0 4. 𝑥2 − 5𝑥 − 14 = 0


Answers:

1. 9(𝑥 + 2) 2. 𝑥(11𝑥 − 6) 3. (𝑤 + 𝑢)(𝑤 − 𝑢) 4. (5𝑥 + 3)(5𝑥 − 3) 5. (8 + 𝑥)(8 − 𝑥) 6. (3𝑚𝑛 + 8𝑝)(3𝑚𝑛 − 8𝑝)

7. (𝑥 + 1)(𝑥 − 1) 8. 4(𝑥 + 4)(𝑥 − 4) 9. 𝑏(𝑎 + 1)(𝑎 − 1) 10. 2(𝑥2 + 9)(𝑥 + 3)(𝑥 − 3) 11. (𝑥 − 5)(𝑥 − 5)

12. (𝑥 + 8)(𝑥 + 2) 13. (𝑥 − 6)(𝑥 + 2) 14. (𝑥 − 12)(𝑥 − 1) 15. (𝑥 + 10)(𝑥 − 3) 16. (𝑥 + 7)(𝑥 + 7) 17. (𝑥 + 9)(𝑥 − 3)

18. (𝑥 + 1)(𝑥 + 1) 19. 3(𝑥 + 3)(𝑥 − 1) 20. 4(𝑥 − 10)(𝑥 + 1) 21. 3(𝑥 + 7)(𝑥 + 1)

𝐵. 1. 𝑥 = 5 2. 𝑥 = 0, 𝑥 = 1

4 3. 𝑥 =

1

2, 𝑥 = −

1

2, 4. 𝑥 = 7, 𝑥 = −2

4.1: EQUIVALENT RATIONAL EXPRESSIONS

Learning Outcomes:

To compare the strategies for writing equivalent forms of rational expressions to

writing equivalent forms of rational numbers

To be able to explain why a given value is non-permissible for a given rational

expression

To determine the non-permissible values for a rational expression

To determine a rational expression that is equivalent to a given rational expression

by multiplying the numerator and denominator by the same factor, and state the

non-permissible values of the equivalent rational expression

A Rational Expression:

an algebraic fraction with a numerator and a denominator that are polynomials.

3𝑥2+12𝑥

3𝑥 is an example of a rational expression

Whenever you are working with algebraic fractions, it is important to determine any

values that must be excluded.

You can write an unlimited number of arithmetic fractions, or rational numbers, of

the form 𝑎

𝑏, where a and b are integers. What integer cannot be used for b?

What happens in each of the following expressions when x = 3?

i) 𝑥−7

𝑥−3 ii)

𝑥−7

𝑥2−9 iii)

𝑥−7

𝑥2−4𝑥+3


What value(s) cannot be used for x in each of the following algebraic functions?

i) 6−𝑥

2𝑥 ii)

3

𝑥−7 iii)

4𝑥−1

(𝑥−3)(2𝑥+1)

Write a rule that explains how to determine any values that a variable cannot be, for any

algebraic fraction.

Note:

Whenever you use a rational expression, you must identify any values that must be

excluded or are considered non-permissible values.

Non-permissible values are all values that make the denominator zero.

Example 1: Determine the non-permissible value(s) for each rational expression:

a. 4𝑎

3𝑏𝑐 b.

𝑥−1

(𝑥+2)(𝑥−3)

c. 11

2𝑥2−6𝑥 d.

𝑥−5

𝑥2−36


Example 2: a. Write a rational number that is equivalent to 8

12

Rational expressions are like fractions. To create an equivalent rational expression,

you can use the same strategy used for rational numbers.

Note, you cannot introduce a factor that required a new restriction, because the

expressions would not have been equivalent.(cannot add any new NPV)

b. Write a rational expression that is equivalent to 4𝑥2+8𝑥

4𝑥

In order to make an equivalent expression, what process do we need to follow?

Example 3: For each of the following, determine if the rational expressions are

equivalent.

a. 9

3𝑥−1 𝑎𝑛𝑑

−18

2−6𝑥 b.

2−2𝑥

4𝑥 𝑎𝑛𝑑

𝑥−1

2𝑥

Substitution can be used to determine if two rational expressions are not equivalent, but it

cannot be used to determine if two rational expressions are equivalent.

Questions page 222-224

#3,5,7,8,9,10,16


4.2 SIMPLIFYING RATIONAL EXPRESSIONS

Learning Outcomes:

To be able to simplify a rational expression

To explain why the non-permissible values of a given rational expression

and its simplified form are the same

To identify and correct errors in a given simplification of a rational

expression, and explain the reasoning

Explore: Use factoring and elimination to write a rational expression that is

equivalent to

−12𝑥3

18𝑥2+18𝑥3

Note:

A rational expression is in simplified form when the numerator and denominator

have no common factors

Always state the non-permissible values of the variables as restrictions before

simplifying a rational expression. Otherwise, if you eliminate a variable or factor

as you simplify, you will lose the information about the non-permissible value of

the variable.

Simplifying rational expressions is similar to simplifying fractions.

Common factors in the numerator and denominator can be cancelled.


Example 1: Simplify

1. 7𝑥6

3𝑥2 , 𝑥 ≠ 0 2. 16𝑥5𝑦 ÷ 12𝑥2𝑦2 , 𝑥 ≠ 0, 𝑦 ≠ 0

3. 15𝑥3𝑦4

−5𝑥𝑦3 , 𝑥 ≠ 0, 𝑦 ≠ 0 4. 5𝑥2−10𝑥

5𝑥 , 𝑥 ≠ 0

5. 18𝑎4+6𝑎3−12𝑎2

12

Example 2: Simplify the following rational expressions:

a. 6𝑚2−8𝑚

3𝑚3−4𝑚2

Steps:

Factor numerator and denominator

Determine NPV

Simplify by cancelling like terms


b. 3𝑎3−3𝑎2

−12𝑎+12

Example 3: Identify and correct the errors in the simplification of each rational

expression.

(a) 2

2 1

x x

x

(b)

3

6x

2x

2

x

x

1

1

1, 1

x

x x

(c) 2

8 12

6 4

x

x x

4 2 3x

2 3 2x x

20,

3

4

2

22 0,

3

x

x

x x


#1-9,13

**careful of negative factors**

Ex.1 2

−2

Ex.2 2−𝑥

𝑥−2

3

62

2 0

x

x x


4.3 MULTIPLYING RATIONAL EXPRESSIONS

Learning Outcomes:

To develop strategies for multiplying rational expressions

To determine the non-permissible values when performing

multiplication on rational expressions

Steps for multiplying rational expressions:

1. Factor numerator and denominator

2. Identify all non-permissible values

3. Simplify using common factors

4. Multiply all numerators together and multiply all denominators together.

Example 1: Simplify (3

10) (

4

9).

Example 2: Simply the following. State the restrictions on the variables.

(A) (15𝑝

7𝑛) (

14𝑛2

5𝑝) (B)

(𝑥+2)

(𝑥−3)(𝑥+5)×

2(𝑥+5)

𝑥(𝑥+2)

Why it is important to identify all non-permissible values before simplifying

a rational expressions


Example 3: Simplify the following products:

a. 2𝑥2−12𝑥

15𝑥∙

5𝑥

𝑥−6 b.

12𝑥3

3𝑥2+6𝑥∙

4𝑥3+8𝑥2

5

c. 𝑥2−9

6𝑥+24∙

10𝑥+40

𝑥(𝑥+3)


#1ac,2ab,3ad,4ac,5a,6,9,

15b


DIVIDING RATIONAL EXPRESSIONS

Learning Outcomes:

To develop strategies for dividing rational expressions

To determine the non-permissible values when performing division on

rational expressions

Investigation

Determine the value of 2

3÷

1

6.

The same strategy is used to divide rational expressions.

Example 1: Simplify: 24𝑦

9𝑥2÷

32𝑦2

18𝑥

non-permissible values when dividing rational expressions:

consider the division 𝑎

𝑏÷

𝑐

𝑑 where a,b,c and d are variables.

The non-permissible values of 𝑎

𝑏 and

𝑐

𝑑 are __________________

The first step to simplifying her is to multiply by the reciprocal 𝑎

𝑏×

𝑑

𝑐.

This introduces another non-permissible value: __________


Steps for dividing rational expressions:

1. Factor numerator and denominator

2. State all restrictions, for 𝑎

𝑏÷

𝑐

𝑑 the NPV occur at b,c and d

3. Take the reciprocal of the second expression

4. Simplify using common factors

5. Multiply all numerators together and multiply all denominators together.

Example 2: Simplify each quotient. State the restrictions on the variable

a. 𝑥3+𝑥2

16÷

𝑥2+𝑥

20𝑥−10

b. 30𝑥2+15𝑥

𝑥−3÷

2𝑥3+𝑥2

𝑥2−3𝑥


c. 4𝑥2−1

𝑥+2÷

4𝑥2+2𝑥

8𝑥2−32

d.

4𝑥+12

3𝑥+15

4𝑥2+12𝑥

(𝑥+5)2


#1bd,2cd,3bc,4bd,5b,7,9,

15a


4.4 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

Learning Outcomes:

To determine, in simplified form, the sum or difference of rational expressions that

have the same denominator

To determine, in simplified form, the sum or difference of rational expressions that

have different denominators

To determine the non-permissible values when performing adding and subtracting

on rational expressions

Reminder: To ADD or SUBTRACT fractions, you need a common denominator

Example 1: Determine each sum or difference. Give answers in lowest terms.

a. 1

8+

5

8 b.

5

6−

3

8

Example 2: The same strategy is used to add or subtract rational expressions.

a. 7𝑥+1

𝑥+

5𝑥−2

𝑥 b.

𝑥2

𝑥−3−

9

𝑥−3 c.

5𝑥+6

6−

2𝑥−9

6

d. 3

8𝑥2+

1

4𝑥


e. 5

𝑥−2−

3

𝑥+2

Steps for adding or subtracting rational expressions:

1. Find a common denominator (may need to factor denominator)

2. Change to equivalent fractions with the common denominator

3. Add or subtract the numerators

4. Reduce if possible

5. State any restrictions (use LCD)

Example 3: Determine each sum or difference. Express each answer in simplest form. Identify all non-permissible values.

a. 10𝑚−1

4𝑚−3−

8−2𝑚

4𝑚−3

b. 𝑥2+2𝑥

(𝑥−3)(𝑥+1)+

3−6𝑥

(𝑥−3)(𝑥+1)−

8

(𝑥−3)(𝑥+1)


Example 4: Determine each sum or difference. Express each answer in

simplest form.

a. 2 1 3 2

6 4

x x

b. 5𝑥+6

3𝑥+9−

𝑚

𝑚+3

c. 2𝑥

𝑥2−1−

4

𝑥−1


#1-9, 13, 14 omit 8a


4.5 SOLVING RATIONAL EQUATIONS

Learning Outcomes:

Identify non-permissible values in a rational equation

Determine the solution to a rational equation algebraically

Examples: Solve the equation:

1. 𝑥

5+

1

10= 2

2. 2

𝑥+

1

3= 4

To eliminate fractions

Multiply both sides of the

equation by the lowest

common denominator


Working with a rational equation is similar to working with rational expressions. A significant difference occurs because in an equation, what you do to one side you must also do to the other side.

Look through the example shown. What steps were used to solve the problem?

2

𝑧2−4+

10

6𝑧+12=

1

𝑧−2

2

(𝑧−2)(𝑧+2)+

10

6(𝑧+2)=

1

𝑧−2

𝑧 ≠ ±2

(𝑧 − 2)(𝑧 + 2)(6) [2

(𝑧−2)(𝑧+2)+

10

6(𝑧+2)] = (𝑧 − 2)(𝑧 + 2)(6) [

1

𝑧−2]

(6)(2) + (𝑧 − 2)(10) = (𝑧 + 2)(6)

12 + 10𝑧 − 20 = 6𝑧 + 12

4𝑧 = 20

𝑧 = 5

2

52 − 4+

10

6(5) + 12=

1

5 − 2

2

21+

10

42=

1

3

What happened during this step?

Step 2?

What was done during this step? Recall how to combine fractions.

Step 4?

Final Step?


Steps: To solve a rational equation:

1. Factor each denominator

2. Identify the non-permissible values

3. Multiply both sides of the equation by the lowest common denominator

4. Solve by isolating the variable on one side of the equation

5. Check your answers

Example 1: Solve the equation. What are the non-permissible values?

2

3 5 2

2 2 4x x x

Check: both sides equal


Example 2: Solve the equation. What are the non-permissible values?

𝑥

𝑥 + 2−

5

𝑥 − 3=

−25

𝑥2 − 𝑥 − 6

Example 3: 18

𝑥2−3𝑥=

6

𝑥−3−

5

𝑥


#1,5,6abd


4.5 APPLICATIONS OF RATIONAL EQUATIONS (Part 2)

Example 1: Two friends share a paper route. Sheena can deliver the papers in 40 min.

Jeff can cover the same route in 50 min. How long, to the nearest minute, does the paper

route take if they work together?

Organize information in a table:

Time to deliver

papers (min)

Fraction of work

done in 1 min

Fraction of work

done in t minutes

Sheena 40 1

40

𝑡

40

Jeff 50 1

50

𝑡

50

Together t 1

𝑡 1

Example 2: Sherry can mow a lawn in 5 hours. Mary can mow the same lawn in 4

hours. Determine how long it would take to mow the lawn if Sherry and Mary worked

together.

Time to

mow

lawn

(hours)

Fraction of

lawn

mowed in 1

hour

Fraction

of lawn

moved in t

hours

Sherry

Mary

Both


Example3: Gerard takes 5 hours longer than Hubert to assemble a play set. If Gerard

and Hubert worked together, they could assemble the play set in 6 hours. Determine how

long it takes each person to assemble the play set if they worked alone.

Time

(hours)

Fraction in

1 hour

Fraction in

t hours

Gerard

Hubert

Both


Example 4: Cameron and Zach work at a garage changing tires on cars. It takes

Cameron 10 minutes longer to change tires on a car. If both of them working together

can change the tires on a car in 12 minutes, algebraically determine how long it takes

each person to change tires if he was working alone.

Time

(hours)

Fraction

in 1 hour

Fraction in

t hours

Cameron

Zach

Both


Example 5: It takes Sheryl twice as long to clean a house as Krista. If they both work

together, they would clean the house in four hours. Determine how long it takes each

person to clean the house individually.

Time

(hours)

Fraction in

1 hour

Fraction in

t hours

Sheryl

Krista

Both


Note: For the remaining questions, the equation will be embedded in the question.

Example 6: A skiing club is going on a skiing trip that costs $1500 total for bussing. If

10 non-members are allowed to go, the price per person drops by $5. If x represents the

number of members and the situation is modelled by 1500 1500

510x x

, algebraically

determine how many members there are.

Example.7: Priddle Inc. is having a Christmas party for all of its employees. Initially, all

employees agree to attend. The cost of catering is $1800, which is to be divided amongst

all people who attend the party. At the last minute, 30 people decide not to come,

increasing the cost per people by $2. If x represents the number of employees and the

situation is modelled by 1800 1800

230x x

, algebraically determine the number of

people who are employed at Priddle Inc.

Questions page 259

#10-12 and application

practice sheet


4.5 Application Practice

1. A large field needs to be mowed. If Amber uses a push mower and Stephanie uses a riding lawn

mower, the lawn can be finished in 4 hours. If Amber uses a push mower only, it takes her 15

hours longer to mow the lawn. Algebraically determine the time it takes to mow the lawn with

the riding mower only.

2. Tyson can wash and dry a sink full of dishes in 20 minutes. It takes Jeffrey three times as long to

wash and dry the same sink full of dishes. Algebraically determine how long it takes to wash

and dry the dishes if they work together.

3. Tamara is an expert at crosswords, but Emily is a novice. It takes Emily an average of 30

minutes longer to complete a medium level puzzle than Tamara. If they both work together, it

takes them a total of 8 minutes. Determine how long it takes each person to complete the

crossword individually.

4. A group of students from Payne Academy are going on a field trip to St. John’s. The bus has a

fixed cost of $2100, which is to be divided equally amongst all the students. To help save

money, a group of 7 students from Regular High School are invited to go, which reduces the cost

per student by $10. If x represents the number of students at Payne Academy and the situation is

modelled by 2100 2100

107x x

, algebraically determine the number of students from Payne

Academy that went on the trip.

5. Rima bought a case of concert T-shirts for $450. She kept two shirts for herself and sold the rest

for $560, making a profit of $10 on each shirt. If x represents the original number of shirts in the

case, and her profit is modelled by the equations 560 450

102x x

, algebraically determine how

many shirts were in the case.

6. When they work together, Stuart and Lucy can deliver flyers to all of the homes in the

neighbourhood in 42 minutes. When Lucy works alone, she can deliver the flyers in 13 minutes

less time than Stuart can when he works alone. Determine how long it takes Stuart to deliver the

flyers when he works alone.

Answers: 1.5 hours 2. 15 min 3. Tamara 10 min Emily 40min 4. 35 students 5. 18 shirts 6. 91min


Chapter 4 Review

1. Find each product or quotient and express in simplest form.

(A) 205

9

3

1642

a

a

a

a (B)

65

1

1

222

aa

a

a

a (C)

2

3

2

2

2 16

4

4

164

x

x

xx

x

x

x

(D) 5

4

25

162

2

x

x

x

x (E)

aa

a

aa

a

2

255

103

522

(F)

4

12

22

yx

yx

2. Find the sum or difference and express in simplest form.

(A) nnn 3

2

62

22

(B) 862

32

22 2

mm

m

m

m

(C) 3

3

65

332

2

xxx

xx (D)

12

1

11

1222

xxx

x

x

3. Completely simplify each of the following complex rational expressions.

(A)

x

x1

1

11

(B)

31

13

x

xx

x

4. Solve. (Be sure to check for extraneous roots.)

(A) 3

32

3

xx

x (B)

1

252

1

3

y

y

y

(C) 2

5

4

32

2

42

xx

x

x (D) 5

1

52

x

x

x

ANSWERS:

1.(a) a5

12(b)

32

12 aa

(c) 4 (d) 5

4

x

x (e)

)5(5 a

a (f)

3

yx 3.(a)

1

1

x

x (b)

3

3

x

x

2. (a) )3(

2

nn

n (b)

)4(2

3

m

m (c)

2

3

x

x (d)

21x

x 4.(a) no solution (b)

2

5 (c) 7 (d)

3

2

Review Factoring Polynomials€¦ · by multiplying the numerator and denominator by the same...

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