Simplifying Expressions 08/09/12lntaylor ©. Table of Contents Learning Objectives Simplifying...

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Simplifying Expressions 08/09/12 lntaylor ©

Transcript of Simplifying Expressions 08/09/12lntaylor ©. Table of Contents Learning Objectives Simplifying...

Page 1: Simplifying Expressions 08/09/12lntaylor ©. Table of Contents Learning Objectives Simplifying Fractions Simplifying Polynomials Simplifying Rational Expressions.

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Simplifying Expressions

08/09/12

Page 2: Simplifying Expressions 08/09/12lntaylor ©. Table of Contents Learning Objectives Simplifying Fractions Simplifying Polynomials Simplifying Rational Expressions.

Table of Contents

Learning Objectives

Simplifying Fractions

Simplifying Polynomials

Simplifying Rational Expressions

The Distributive Property

Practice

1

2

3

4

5

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Learning Objectives

LO 1 Understand the difference between expressions and equations

TOC

LO 2 Correctly simplify expressions containing fractions and exponents

LO 3 Correctly use the principle of CLT – combine like terms

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Definitions

Definition 1 Expressions do not contain = ≠ < > ≤ ≥

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Definition 2 Fractions are rational numbers consistingof a numerator and denominator i.e. ¼ , ½ , ¾

Definition 3 Terms are numbers, letters and exponents, or a combination ofthese things, separated by an operand symbol (+, −, ∗, ÷)

Example

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2x + 3 where 2x and 3 are both terms3x2÷ x where 3x2 and x are both terms

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Previous Knowledge

PK 1

PK 2

Basic Operations and Properties

Fractions

PK 3 Combining Like Terms

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PK 4 Exponent Rules

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Simplifying Fractions

Note1 The following is a review of the Fractions PowerPoint

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Rule 1

Rule 2

Adding and subtracting fractions requires cross multiplication

Multiplying fractions requires straight across multiplication

Rule 3 Dividing requires flipping a fraction and multiplying straight across

Rule 4 Learn to “get rid” of fractions by turning expressions into equations

Basic Rules of Fractions

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Adding Fractions

2 + 3 5 7

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Step 1

Step 2

Construct matrix with numerators on top and denominators on side

Blank out boxes diagonally

Step 3 Multiply matrix

Step 4 Add the results; this becomes the numerator

2 + 35 7

+ 15

+ 14

= 29

5 x 7 = 35

2 35 7

Step 5 Multiply left side numbers (denominators); this becomes the denominator

35

Step 6 Reduce fraction if possible

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Now you try

3 + 54 7

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Step 1

Step 2

Construct matrix with numerators on top and denominators on side

Blank out boxes diagonally

Step 3 Multiply matrix

Step 4 Add the results; this becomes the numerator

3 + 54 7

+ 20

+ 21

= 41

4 x 7 = 28

3 54 7

Step 5 Multiply left side numbers (denominators); this becomes the denominator

28

Step 6 Reduce fraction if possible

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Is there another method?

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Alternate Method

3 + 5 - 1 4 7 6

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Step 1

Step 2

Multiply every numerator by every other denominator

Add the results; this is your numerator

Step 3 Multiply the denominators; this is your denominator

Step 4

3 + 54 7

Reduce fraction if possible

─ 16

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37 6

3 x 7 x 6 = 1264

56

5 x 4 x 6 = 120

- 1

-1 x 7 x 4 = - 28

218

4 x 7 x 6 = 168

___168

Step 5 The easy way to reduce fractions is… Subtract the numerator and denominator…Do this until the result is less than the denominator and reduce

218–168 = 50218 = 1 + 50168 168

= 1 + 25 84

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Now you try!

3 + 5 + 1 5 7 3

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Step 1

Step 2

Multiply every numerator by every other denominator

Add the results; this is your numerator

Step 3 Multiply the denominators; this is your denominator

Step 4

3 + 55 7

Reduce fraction if possible

+ 13

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37 3

3 x 7 x 3 = 635

53

5 x 5 x 3 = 75

1

1 x 7 x 5 = 35

173

5 x 7 x 3 = 105

___105

Step 5 The easy way to reduce fractions is… Subtract the numerator and denominator…Do this until the result is less than the denominator and reduce

173–105 = 68173 = 1 + 68105 105

= 1 + 68 105

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Rule 1

Rule 2

Adding and subtracting fractions requires cross multiplication

Multiplying fractions requires straight across multiplication

Rule 3 Dividing requires flipping a fraction and multiplying straight across

Rule 4 Learn to “get rid” of fractions by turning expressions into equations

Basic Rules of Fractions

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Rule 1

Rule 2

Multiply numerators; this becomes the new numerator

Multiply denominators; this becomes the new denominator

Rule 3 Reduce fraction if possible

23

57

2 (5) = 103 7 21

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Now you try!

34 43

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Rule 1

Rule 2

Multiply numerators; this becomes the new numerator

Multiply denominators; this becomes the new denominator

Rule 3 Reduce fraction if possible

34

34

3 (3) = 94 4 16

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Understanding Cross Cancellation

74 63

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Rule 1

Rule 2

Numerators can be moved anytime YOU want

Reduce fraction

Rule 3 Multiply straight across

34

76

3 (7)(4) 6

12

1 x 7 = 72 x 4 = 8

Rule 4 Reduce fraction if possible

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Now you try!

54 93

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Rule 1

Rule 2

Numerators can be moved anytime YOU want

Reduce fraction

Rule 3 Multiply straight across

34

59

3 (5)(4) 9

13

1 x 5 = 53 x 4 = 12

Rule 4 Reduce fraction if possible

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Dividing Fractions

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Rule 1

Rule 2

Adding and subtracting fractions requires cross multiplication

Multiplying fractions requires straight across multiplication

Rule 3 Dividing requires flipping a fraction and multiplying straight across

Rule 4 Learn to “get rid” of fractions by turning expressions into equations

Basic Rules of Fractions

TOC08/09/12

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Divide

54 93 /

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Rule 1

Rule 2

Write top fraction

Flip bottom fraction

Rule 3 Check for cross cancellation; you can here but we will skip it

34

95

Rule 4 Multiply straight across

34

95

3 x 5 = 154 x 9 = 36

Rule 5 Reduce fraction if possible

512

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Now you try!

45 73 /

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Rule 1

Rule 2

Write top fraction

Flip bottom fraction

Rule 3 Check for cross cancellation; none here

35

47

Rule 4 Multiply straight across

35

47

3 x 7 = 215 x 4 = 40

Rule 5 Reduce fraction if possible

2140

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Simplifying Polynomials

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Simplify a Polynomial Expression

3x2 + 3x + 3 + x2 – 2x – 2

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+ x23x2

Step 1

Step 2

4x2

+ 3x + 3 – 2x – 2

+ x + 1

Look for the same variable and exponent combinations

Combine like terms in columns

Step 3 Add terms

Note: When you add or subtract polynomials exponents do not change

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Now you try

10x2 – 7x + 18 – 3x2 – 3x – 7

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– 3x210x2

Step 1

Step 2

7x2

– 7x + 18 – 3x – 7

–10 x + 11

Look for the same variable and exponent combinations

Combine like terms in columns

Step 3 Add terms

Note: When you add or subtract polynomials exponents do not change

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Simplifying Rational Expressions

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Simplify

2x2 + 4x – 10x 3 5

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Step 4

Step 5

Combine like terms if necessary

Divide by the y coefficient

Step 6 Simplify if possible

Step 7 You can erase the “= y ” if you want

Step 2

Step 1 Turn the expression into an equation by introducing “ = y”

Every term gets a denominator

Step 3 Multiply every term’s numerator with every other denominatorThen multiply the denominators

2x²3

+ 4x – 10x15

= y(5) (1)

2x² + 4x(3) (1)

– 10x(3) (5) (1)(3) (5)

=

y

10x² + 12x – 150x = 15y

10x² – 138x = 15y

10x² – 138x = y 15

x (10x – 138) 15

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Now you try!

2x2 + 3x – 10x 7 5

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Step 4

Step 5

Combine like terms if necessary

Divide by the y coefficient

Step 6 Simplify if possible

Step 7 You can erase the “= y ” if you want

Step 2

Step 1 Turn the expression into an equation by introducing “ = y”

Every term gets a denominator

Step 3 Multiply every term’s numerator with every other denominatorThen multiply the denominators

2x²7

+ 3x – 10x15

= y(5) (1)

2x² + 3x(7) (1)

– 10x(7) (5) (1)(7) (5)

=

y

10x² + 21x – 350x = 35y

10x² – 329x = 35y

10x² – 329x = y 35

x (10x – 329) 35

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The Distributive Property

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Step 4

Step 5

Look for the same variable/exponent combinations (none here)

Combine any like terms in columns (none here)

Note: When you add or subtract polynomials exponents do not change

Step 2

Step 1 The Distributive Property means multiply the term outside the ( )

Multiply coefficients and watch your signs

3∗5x

+ 3∗7

Step 3 Rewrite with one sign for each term (not needed here)

3(5x + 7)

TOC08/09/12

15x + 21

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Now you try!

5(4x - 9)

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Step 4

Step 5

Look for the same variable/exponent combinations (none here)

Combine any like terms in columns (none here)

Note: When you add or subtract polynomials exponents do not change

Step 2

Step 1 The Distributive Property means multiply the term outside the ( )

Multiply coefficients and watch your signs

5∗4x

+ 5∗-9

Step 3 Rewrite with one sign for each term (not needed here)

5(4x - 9)

TOC08/09/12

20x - 45

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Now you try!

5(4x2 – 9x + 10)

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Step 4

Step 5

Look for the same variable/exponent combinations (none here)

Combine any like terms in columns (none here)

Note: When you add or subtract polynomials exponents do not change

Step 2

Step 1 The Distributive Property means multiply the term outside the ( )

Multiply coefficients and watch your signs

5∗4x2+ 5∗-9x

Step 3 Rewrite with one sign for each term (not needed here)

5(4x2 – 9x + 10)

TOC08/09/12

20x2 – 45x + 50

+ 5∗10

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Now try something harder!

– 20x2 +10x – 18 – 3 (– 5x2 + 3x – 7)

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– 20x2

Step 4

Step 5

– 5x2

+ 10x – 18

+ x + 3

Look for the same variable and exponent combinations

Combine like terms in columns

Step 6 Add terms

Note: When you add or subtract polynomials exponents do not change

Step 2

Step 1 – ( ) means a red flag – mistake zone

Multiply coefficients and then add the – to each sign in the ( )

– – 15x2 – + 9x – – 21

Step 3 Rewrite with one sign for each term

+ 15x2 – 9x + 21

– 3(– 5x2 + 3x – 7)

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Now you try

– 2x2 + 4x – 10 – 2(4x2 + 2x – 6)

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– 2x2

Step 4

Step 5

– 10x2

+ 4x – 10

+ 2

Look for the same variable and exponent combinations

Combine like terms in columns

Step 6 Add terms

Note: When you add or subtract polynomials exponents do not change

Step 2

Step 1 – ( ) means a red flag – mistake zone

Multiply coefficients and then add the – to each sign in the ( )

– + 8x2 – + 4x – – 12

Step 3 Rewrite with one sign for each term

– 8x2 – 4x + 12

– 2(4x2 + 2x – 6)

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Practice

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Problem Answer

Simplify 3/8 + 1/3 – 2/5

Simplify 2/3 – 5/6 + 1/2

Simplify 2x + 13 – 4x – 10

Simplify 2(– 3x – 7)

Simplify 3x2(-3x – 7)

Simplify – 2x(– 3x + 8) – (2x + 9)

Simplify 2(3/8 – 2/9)

Simplify 14x2 + 8x – 9 + 8x3 – 4x2 + 8

Simplify (2/3 + 1/9 – 1/3)2

> 37/120

>

>

>

>

>

1/3

– 2x + 3

– 6x – 14

– 9x3 – 21x2

6x2 – 18x – 9

> 11/36

> 8x3 + 10x2 + 8x – 1

> 16/81

clear answers