13 multiplication and division of rational expressions

Multiplication and Division of Rational Expressions

Multiplication Rule for Rational Expressions A B

In most problems, we reduce the product by factoring the top and the bottom, then cancel.

Example A. Simplify

10xy3z

a. * y2

Example A. Simplify

10xy3z

a. * y2

5x3y3z

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

(x + 3)(x – 1 )

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

(x + 3)(x – 1 ) (x + 2 )(x – 2)

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

(x + 3)(x – 1 ) (x – 2)

(x + 2 )(x – 2)

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

(x + 3)(x – 1 ) (x – 2) x(x – 1 )

(x + 2 )(x – 2)

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

(x + 3)(x – 1 ) (x – 2) x(x – 1 )

(x + 2 )(x – 2)

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )*

(x + 3)(x – 1 ) (x – 2) x(x – 1 )

(x + 2 )(x – 2)

Example A. Simplify

10xy3z

a. * y2

5x3y3z =

b. (x2 + 2x – 3 ) (x – 2) (x2 – x )

(x2 – 4 )

=(x + 3)(x + 2)

(x + 3)(x – 1 ) (x – 2) x(x – 1 )

(x + 2 )(x – 2)

In the next section, we meet the following type of problems.

Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

Example B. Simplify and expand the answers.

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) – x + 1

(x – 3)(x + 1)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)

(x – 2)(x + 1) – (x + 1)(x + 3)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)

(x – 2)(x + 1) – (x + 1)(x + 3)

= (x – 2)(x + 1) + (–x –1)(x + 3)

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)

(x – 2)(x + 1) – (x + 1)(x + 3)

= (x – 2)(x + 1) + (–x –1)(x + 3)

= x2 – x – 2 – x2 – 4x – 3

a. x + 3 x – 1 (x2 – 1)

x + 3 (x – 1) (x – 1)(x + 1)

(x + 3)(x + 1) = x2 + 4x + 3

b. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

x – 2 (x – 3)(x + 3) [ – x + 1

(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)

(x – 2)(x + 1) – (x + 1)(x + 3)

= (x – 2)(x + 1) + (–x –1)(x + 3)

= x2 – x – 2 – x2 – 4x – 3

= –5x – 5

Division Rule for Rational Expressions

÷ = ADBC

Reciprocate

÷ = ADBC

Reciprocate

We convert division by an expression of multiplying by its reciprocal.

÷ = ADBC

Reciprocate

We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

Example C. Simplify

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

Example C. Simplify

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

=2(x – 3) (x + 3)

Example C. Simplify

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

=2(x – 3) (x + 3)

(x + 3)(x – 1) (3 – x)(3 + x)

Example C. Simplify

÷ = ADBC

Reciprocate

Example C. Simplify

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

=2(x – 3) (x + 3)

(x + 3)(x – 1) (3 – x)(3 + x)

(9 – x2)

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

=2(x – 3) (x + 3)

(x + 3)(x – 1) (3 – x)(3 + x)

(–1)

Example C. Simplify

÷ = ADBC

Reciprocate

(2x – 6) (x + 3)

÷(x2 + 2x – 3)

(9 – x2)

=(2x – 6) (x + 3)

(x2 + 2x – 3) (9 – x2)

=2(x – 3) (x + 3)

(x + 3)(x – 1) (3 – x)(3 + x)

(–1)

=–2(x – 1)

(3 + x)

Example C. Simplify

Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences.

Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this.

Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.

Example D. Break up the numerators as the sums or differences and simplify each term.

(2x – 6) (x + 3)

Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.

(2x – 6) 3x2

(2x – 6) (x + 3)

(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

(2x – 6) (x + 3)

(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2

(2x – 6) (x + 3)

(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2 = – 2

(2x – 6) (x + 3)

(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2 = – 2

3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable.

(2x – 6) (x + 3)

(x + 3) – (x + 3)2x 6

(2x – 6) 3x2

b. = – 2x 63x2 3x2 = – 2

3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable. Specifically, long division gives relevant results only when the degree of the numerator is the same or more than the degree of the denominator.

Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.

Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient.

Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125

ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.

Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125

iii. Repeat steps i and ii until no more quotient may be entered.

i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125

Let’s look at the example 125/8 or 125 ÷ 8 by long division.

iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R

Dand that R (the remainder) is smaller then D (no more quotient).

where Q is the quotient

Dand that R (the remainder) is smaller then D (no more quotient).

1258 = 15 + 5

Multiplication and Division of Rational ExpressionsExample E. Divide using long division(2x – 6)

(x + 3)

i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.

Example E. Divide using long division(2x – 6) (x + 3)

)x + 3 2x – 6

Make sure the terms are in order.

)x + 3 2x – 6

enter the quotients of the leading terms 2x/x = 2

)x + 3 2x – 6 2

enter the quotients of the leading terms 2x/x = 2

)x + 3 2x – 6 2

2x + 6ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.

)x + 3 2x – 6 2

2x + 6–12

)x + 3 2x – 6 2

2x + 6–12

)x + 3 2x – 6 2

2x + 6–12

Stop. No more quotient since x can’t going into 12.iii. Repeat steps i and ii until no more

quotient may be entered.

)x + 3 2x – 6 2

2x + 6–12

iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:

ND = Q + R

Dhas smaller degree then denominator D (no more quotient).

where Q is the quotient and the remainder R

)x + 3 2x – 6 2

2x + 6–12

Hence we may write(2x – 6) (x + 3)

ND = Q + R

)x + 3 2x – 6 2

2x + 6–12

ND = Q + R

)x + 3 2x – 6 2

2x + 6–12

= 2 – 12 x + 3

ND = Q + R

Multiplication and Division of Rational ExpressionsExample F. Divide using long division

x2 – 6x + 3x – 2

ND = Q + R

)x + 3

x2 – 6x + 3

ND = Q + R

Example F. Divide using long divisionx2 – 6x + 3x – 2

)x + 3

x2 – 6x + 3

ND = Q + R

)x + 3 x

x2 – 6x + 3

ND = Q + R

)x + 3 x

x2 + 3x

x2 – 6x + 3

ND = Q + R

)x + 3 x

x2 + 3x–9x + 3

–)x2 – 6x + 3

ND = Q + R

)x + 3 x – 9

x2 + 3x–9x + 3

–)x2 – 6x + 3

ND = Q + R

)x + 3 x – 9

x2 + 3x–9x + 3

–)x2 – 6x + 3

–9x – 27

ND = Q + R

)x + 3 x – 9

x2 + 3x–9x + 3

–)x2 – 6x + 3

–9x – 27–)30

ND = Q + R

)x + 3 x – 9

x2 + 3x–9x + 3

–)x2 – 6x + 3

–9x – 27–)30

Stop. No more quotient since x can’t going into 30. Hence 30 is the remainder.

and the remainder R

)x + 3 x – 9

x2 + 3x–9x + 3

–)x2 – 6x + 3

–9x – 27–)30

x2 – 6x + 3x – 2

ND = Q + R

)x + 3 x – 9

x2 + 3x–9x + 3

–)x2 – 6x + 3

–9x – 27–)30

x2 – 6x + 3x – 2

= x – 9 + 30x + 3

ND = Q + R

Ex A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions

1. 10x * 25x3

* 1625x4

10x* 35x32.

5. 109x4

* 185x3

3.12x6* 5

27* 63

10x* 35x34.

7. 75x49

* 4225x3

9.2x – 4 2x + 4

5x + 10 3x – 6

10.6 – 4x 3x – 2

x – 2 2x + 4

11.9x – 12 2x – 4

2 – x 8 – 6x

12. x + 4

–x – 44 – x

x – 4

13.3x – 9

15x – 53 – x

5 – 15x14.

42 – 6x –2x + 14

4 – 2x –7x + 14

15.(x2 + x – 2 ) (x – 2) (x2 – x)

(x2 – 4 )*

16. (x2 + 2x – 3 ) (x2 – 9) (x2 – x – 2 )

(x2 – 2x – 3)*

17.(x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1)

(x2 + x )*

18. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 )

(x2 – 5x + 6)*

19.(x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8)

(x2 – 3x + 2)*

20.(– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12 )

(6 – x2 – x)*

Ex. A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions

21. (2x2 + x – 1 ) (1 – 2x)

(4x2 – 1) (2x2 – x )

22. (3x2 – 2x – 1) (1 – 9x2)

(x2 + x – 2 ) (x2 + 4x + 4)

23.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1)

(–x – 3x2)24. (x + 1 – 6x2)

(–x2 – 4)(2x2 + x – 1 ) (x2 – 5x – 6)

25. (x3 – 4x) (–x2 + 4x – 4)

(x2 + 2) (–x + 2)

26. (–x3 + 9x ) (x2 + 6x + 9)(x2 + 3x) (–3x2 – 9x)

Ex. B. Multiply, expand and simplify the results.

27. x + 3 x + 1 (x2 – 1) 28. x – 3

x – 2 (x2 – 4) 29. 2x + 3 1 – x (x2 – 1)

30. 3 – 2x x + 2 (x + 2)(x +1) 31. 3 – 2x

2x – 1 (3x + 2)(1 – 2x)

32. x – 2 x – 3 ( + x + 1

x + 3 )( x – 3)(x + 3)

33. 2x – 1 x + 2 ( – x + 2

2x – 3 ) ( 2x – 3)(x + 2)

38. x – 2 x2 – 9 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)

39. x + 3 x2 – 4 ( – 2x + 1

x2 + x – 2 ) ( x – 2)(x + 2)(x – 1)

40. x – 1 x2 – x – 6 ( – x + 1

x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1)

41. x + 2 x2 – 4x +3( – 2x + 1

x2 + 2x – 3 ) ( x – 3)(x + 3)(x – 1)

34. 4 – x x – 3 ( – x – 1

2x + 3 )( x – 3)(2x + 3)

35. 3 – x x + 2 ( – 2x + 3

x – 3 )(x – 3)(x + 2)

Ex B. Multiply, expand and simplify the results.

36. 3 – 4x x + 1 ( – 1 – 2x

x + 3 )( x + 3)(x + 1)

37. 5x – 7 x + 5 ( – 4 – 5x

x – 3 )(x – 3)(x + 5)

Ex. C. Break up the following expressions as sums and differences of fractions.

42. 43. 44.

45. 46. 47.

x2 + 4x – 6 2x2x2 – 4

12x3 – 9x2 + 6x3x

x2 – 4 2x

xx8 – x6 – x4

x2x8 – x6 – x4

Ex D. Use long division and write each rational expression in

the form of Q + .RD

(x2 + x – 2 ) (x – 1)

(3x2 – 3x – 2 ) (x + 2)

2x + 6 x + 2 48. 3x – 5

x – 2 49. 4x + 3 x – 1 50.

5x – 4 x – 3 51. 3x + 8

2 – x52. –4x – 5 1 – x53.

54. (2x2 + x – 3 ) (x – 2)

55. 56.

(–x2 + 4x – 3 ) (x – 3)

(5x2 – 1 ) (x – 4)

57. (4x2 + 2 ) (x + 3)

58. 59.

13 multiplication and division of rational expressions

Education

Lesson 28: Rational Expressions and Equations...Lesson 28 Activity 1: Simplifying Rational Expressions Time: 20-25 Minutes 1. Rational expressions are like rational numbers (can’t

Objective SWBAT simplify rational expressions, add, subtract, multiply, and divide rational expressions and solve rational equations SWBAT simplify rational

Chapter 6 Rational Expressions § 6.1 Rational Functions and Multiplying and Dividing Rational Expressions

Unit 4 - Rational Expressions Notesvalentinesclass.weebly.com/.../unit_4_-_rational_expressions_notes.pdf · 4.3 Add & Subtract Rational Expressions • Using Rational Expressions

Rational Expressions Simplifying Rational Expressions

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS: MULTIPLICATION: COMBINE RIGHT ACROSS THE MULTIPLICATION SIGN. DIVISION: FLIP THE SECOND RATIONAL AND MULTIPLY

Chapter 6 Rational Expressions and Equations Section 6.1 Multiplying Rational Expressions

Rational Expressions Handout

6.2 – Rational Expressions

Rational Algebraic Expressions Products and Quotients of Rational Expressions Sums and Differences of Rational Expressions Complex Fractions Fractional Coefficients Fractional Equations

Adding and Subtracting Rational Expressions - Cobb … · Adding and Subtracting Rational Expressions ... Lesson Presentation Lesson Quiz. ... Rational Expressions Warm Up Add or

Multiply and Divide Rational Expressionslehimath.weebly.com/.../6.1_multiply,_divide_rational_expressions.pdf · Rational Expressions - Simplify Rational expressions can be simplified

Rational Algebraic Expressions Objective: To simplify rational algebraic expressions

Rational Expressions. Martin-Gay, Developmental Mathematics 2 Rational Expressions Examples of Rational Expressions

RATIONAL EXPRESSIONS. Rational Expressions and Functions: Multiplying and Dividing Objectives –Simplifying Rational Expressions and Functions –Rational

1 Topic 8.2.1 Multiplying Rational Expressions Multiplying Rational Expressions

Rational Expressions - Gavilan College -> Faculty / …hhh.gavilan.edu/ybutterworth/intermediate/ch6Angel.doc · Web view6.1 The Domains of Rational Expressions & Multiplication &

Simplifying Rational Expressions

Chapter 6 RATIONAL EXPRESSIONS AND EQUATIONS. Chapter 6 6.1 – RATIONAL EXPRESSIONS

Rational Expressions and Equations · Rational Numbers Examples: Equivalent Rational Numbers: Multiplying/Dividing Rational Numbers: Factor, reduce, then multiply. Rational Expressions