11

OCF.01.9 - Multiplying and OCF.01.9 - Multiplying and Dividing Rational ExpressionsDividing Rational Expressions

MCR3U - SantowskiMCR3U - Santowski

22

(A) Review of Factoring - Common Factoring(A) Review of Factoring - Common Factoring

when we when we expandexpand, we multiply two (or more) polynomials together to come up with a , we multiply two (or more) polynomials together to come up with a single polynomialsingle polynomial

when we when we factorfactor, we do the opposite - we go from a single polynomial to a product of a , we do the opposite - we go from a single polynomial to a product of a pair (or more) of polynomialspair (or more) of polynomials

   to to factorfactor means to find numbers that multiply together to give a product means to find numbers that multiply together to give a product ex. factors of ex. factors of

24 are 2x12 i.e. (2)(12), 1x24, 3x8, 4x624 are 2x12 i.e. (2)(12), 1x24, 3x8, 4x6

the the greatest common factorgreatest common factor is the largest factor of a set of given number is the largest factor of a set of given number ex. GCF ex. GCF of 24 and 36 is 12 although 2,3,4,6 are all common factorsof 24 and 36 is 12 although 2,3,4,6 are all common factors ex. GCF of 8xex. GCF of 8x22y and 12axyy and 12axy22 is 4xyis 4xy

when we factor algebraic expressions, the first step is to find and GCF for all the termswhen we factor algebraic expressions, the first step is to find and GCF for all the terms ex. factor 24xex. factor 24x22 + 6x + 6x ex. factor 5aex. factor 5a22 - 15ab - 15ab22 + 30a + 30a22bb22

ex. factor and check by expanding 3xex. factor and check by expanding 3x22 - 9x + 15x - 9x + 15x22yy   ex. factor 5(x + 3) + 2x(x + 3)ex. factor 5(x + 3) + 2x(x + 3)

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(B) Review of Factoring - Factoring Simple Trinomials(B) Review of Factoring - Factoring Simple Trinomials

find a pair of numbers that satisfy the given set of conditions:find a pair of numbers that satisfy the given set of conditions: multiply to 10 and add to 7multiply to 10 and add to 7 multiply to 3 and add to 4multiply to 3 and add to 4 multiply to 5 and add to 6multiply to 5 and add to 6 multiply to -8 and add to 2multiply to -8 and add to 2 multiply to -8 and add to –2multiply to -8 and add to –2 multiply to -12 and add to –1multiply to -12 and add to –1 multiply to -12 and add to 1multiply to -12 and add to 1

factor xfactor x22 + 2x + 1 + 2x + 1 factor xfactor x22 - 4x – 3 - 4x – 3 factor afactor a22 + 3a – 40 + 3a – 40 factor dfactor d22 + 12d – 160 + 12d – 160 factor gfactor g22 - 2g + 36 - 2g + 36

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(C) Review of Factoring - Factoring Trinomials When a(C) Review of Factoring - Factoring Trinomials When a11

Recall the expansion using FOILRecall the expansion using FOIL ex. (2b - 5)(4b + 3) ex. (2b - 5)(4b + 3) = 2b x (4b + 3) – 5 x (4b + 3)= 2b x (4b + 3) – 5 x (4b + 3) = (2b)(4b) + (2b)(3) + (-5)(4b) + (-5)(3)= (2b)(4b) + (2b)(3) + (-5)(4b) + (-5)(3) = 8b= 8b22 + 6b - 20b - 15 + 6b - 20b - 15 = 8b= 8b22 - 14b - 15 - 14b - 15

observe that the -14b term came from two terms, 6b and -20b and observe that the product of (-observe that the -14b term came from two terms, 6b and -20b and observe that the product of (-20)(6) = the product of (8)(-15) = -12020)(6) = the product of (8)(-15) = -120

so, to factor 8bso, to factor 8b22 - 14b -15, we work through the reverse procedure. - 14b -15, we work through the reverse procedure. 8b8b22 - 14b - 15 - 14b - 15 = 8b= 8b22 + 6b - 20b - 15 + 6b - 20b - 15 = 2b(4b + 3) -5(4b + 3)= 2b(4b + 3) -5(4b + 3) = (4b + 3)(2b - 5)= (4b + 3)(2b - 5)   In the decomposition method, we decompose the middle term into two terms and then proceed In the decomposition method, we decompose the middle term into two terms and then proceed

using factoring by grouping.using factoring by grouping.   ex. Factor 6xex. Factor 6x22 - 17x + 10 - 17x + 10 ex. Factor 6xex. Factor 6x33 – 32x – 32x22 + 10x + 10x

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(D) Review of Multiplying and Dividing With Fractions(D) Review of Multiplying and Dividing With Fractions

recall how to divide and multiply rational numbers - fractionsrecall how to divide and multiply rational numbers - fractions

ex. (6/5) x (25/42) - multiply the numerator and then multiply the ex. (6/5) x (25/42) - multiply the numerator and then multiply the denominator denominator (150/210) = 5/7 (150/210) = 5/7

OR look to cancel COMMON FACTORS between the numerator OR look to cancel COMMON FACTORS between the numerator and denominators prior to multiplying and denominators prior to multiplying

(6/5) x (5x5)/(6x7) = (1/1) x (5/7) = 5/7(6/5) x (5x5)/(6x7) = (1/1) x (5/7) = 5/7

ex. (5/12) ex. (5/12) (15/64) - multiply by the reciprocal (5/12) x (64/15) (15/64) - multiply by the reciprocal (5/12) x (64/15) (5/4x3) ÷ (4x16/3x5) (5/4x3) ÷ (4x16/3x5) (16/3x3) (16/3x3) 16/9 16/9

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(E) Multiplying and Dividing Rational Expression2 – Example 1(E) Multiplying and Dividing Rational Expression2 – Example 1

the same strategy that was introduced yesterday will be employed again - the same strategy that was introduced yesterday will be employed again - we must try to factor the expressions in the numerator and denominatorwe must try to factor the expressions in the numerator and denominator

Example: Simplify Example: Simplify 7

3

21

5

2

2

ab

c

c

a

7

3

21

57

1 3

3 7

7

1

7

549

50 0

2

2

ab

c

c

aa b

c

c c

a a

b c

abc

aa c

( )

( )

( )

(5 )

; ;

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(E) Multiplying and Dividing Rational Expressions – Example 2(E) Multiplying and Dividing Rational Expressions – Example 2

Simplify Simplify y

a

y

a

2

3

2 4

9

y

a

y

ay

a

a

ya y

a y

2

3

2 4

91 2

3

3 3

2 20 2

3

20 2

( ) ( )

( ); ;

; ;

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(E) Multiplying and Dividing Rational Expressions – Example 3(E) Multiplying and Dividing Rational Expressions – Example 3

Simplify Simplify a a

a a

a a

a a

2

2

2

2

2 3

6 5

7 10

6

a a

a a

a a

a aa a

a a

a a

a aa

a

a

2

2

2

2

2 3

6 5

7 10

63 1

5 1

5 2

3 25 1 3 2

1

15 1 3 2

1 5 1 3 2

( )( )

( )( )

( )( )

( )( ); , , ,

; , , ,

; , , ,

99

(E) Multiplying and Dividing Rational Expressions – Example 4(E) Multiplying and Dividing Rational Expressions – Example 4

Simplify Simplify 2 3 2

3 8 4

2 3 1

3 2

2

2

2

2

d d

d d

d d

d d

2 3 2

3 8 4

2 3 1

3 22 1 2

3 2 2

3 2

1 2 1

11 2

2

3

1

20

2

2

2

2

d d

d d

d d

d dd d

d d

d d

d d

d

dd

( )( )

( )( )

( )

( )( )

; , , , ,

1010

(F) Internet Links(F) Internet Links

College Algebra Tutorial on Multiplying anCollege Algebra Tutorial on Multiplying and Dividing Rational Expressions from Westd Dividing Rational Expressions from West Texas A&M Texas A&M

Multiplying & Dividing Multiplying & Dividing RationalsRationals Lesson - I from Purple Math Lesson - I from Purple Math

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(G) Homework(G) Homework

Page 106, Q5,9,12,15,16Page 106, Q5,9,12,15,16Do eol for each questionDo eol for each question

Nelson Text, p359, Q1-5 eol, 7-9 eol, 12, Nelson Text, p359, Q1-5 eol, 7-9 eol, 12, 1717