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Simplifying Expressions
08/09/12
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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6
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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 = ≠ < > ≤ ≥
TOC
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
TOC08/09/12
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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
TOC08/09/12
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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
TOC08/09/12
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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)
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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)
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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)
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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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08/09/12 lntaylor ©TOC
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