copyright © 2009 pearson education, inc. publishing as pearson addison-wesley rules for exponents...
TRANSCRIPT
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Rules for Exponents
Review of Bases and Exponents
Zero Exponents
The Product Rule
Power Rules
5.1
Slide 3Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Review of Bases and Exponents
The expression 53 is an exponential expression with base 5 and exponent 3.
Its value is 5 5 5 = 125.
bn
Base
Exponent
times
...n
b b b b
Slide 4Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Evaluating exponential expressions
Evaluate each expression. a. b. c.
Solutiona. b. c.
242
8
43 4( 3)
242
8
2 factors
28
4 4
162
8
2 2 4
434 factors
3 3 3 3( )
81
4 factors
( 3) ( 3) ( 3) ( 3)
81
4( 3)
Slide 5Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Zero Exponents
Slide 6Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Evaluating exponential expressions
Evaluate each expression. Assume that all variables represent nonzero numbers. a. b. c.
Solution
a. b. c.
08
02
43
03 7
2
x y
z
080
24
3
03 7
2
x y
z
1
4(1) 4
1
Slide 7Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Product Rule
Slide 8Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using the product rule
Multiply and simplify. a. b. c.
Solution
a. b. c.
2 43 32 73 6x x 2 2(3 4 )x x x
2 43 3 2 73 6x x 2 2(3 4 )x x x2 43 3
2 73 6 x x
2 43 63
729
2 718x
918x
2 2 23 4x x x x
3 43 4x x
Slide 9Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponent Rules
Slide 10Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Raising a power to a power
Simplify the expression. a. b.
Solution
a. b.
323 52x
323 52x
2 33 2 5x
6310x
Slide 11Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponent Rules
Slide 12Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Raising a product to a power
Simplify the expression. a. b. c.
Solution
a. b. c.
3(2 )a 2 3( 3 )x 3 4 2( 2 )h
3(2 )a 2 3( 3 )x 3 4 2( 2 )h3(2 )a
3 32 a
38a
2 3( 3 )x
3 2 3( 3) ( )x
627x
4 2( 8 )h 2 4 2( 8) ( )h
864h
Slide 13Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponent Rules
Slide 14Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Raising a quotient to a power
Simplify the expression. a. b. c.
Solution
a. b. c.
33
4
7a
b
3
4
x y
33
4
7a
b
3
3
( )
4
x y
3
3
3 27
4 64
7
7
a
b
3
4
x y
3( )
64
x y
Slide 15Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Combining rules for exponents
Simplify the expression. a. b. c.
Solution
a. b. c.
2 3(3 ) (4 )a a
32 4a b
d
2 2 3 4 3(3 ) ( 5 )a b a b
2 32 3 3 32 2 43 ( ) ( 5) ( ) ( )a b a b 2 2 3 33 4a a 2 3 4 3
3
( ) ( )a b
d
2 3(3 ) (4 )a a
32 4a b
d
2 2 3 4 3(3 ) ( 5 )a b a b
6 12
3
a b
d
4 2 9 129 ( 125)a b a b 4 9 2 129( 125)a a b b
13 141125a b
2 39 64a a 5576a
Slide 16Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Addition and Subtraction of Polynomials
Monomials and Polynomials
Addition of Polynomials
Subtraction of Polynomials
Evaluating Polynomial Expressions
5.2
A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers.Examples of monomials: The degree of monomial is the sum of the exponents of the variables. If the monomial has only one variable, its degree is the exponent of that variable.
Slide 18Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
3 2 9 88, 7 , , 8 , y x x y xy
The number in a monomial is called the coefficient of the monomial.
Slide 19Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Identifying properties of polynomials
Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains and its degree.
a. 9y2 + 7y + 4 b. 7x4 – 2x3y2 + xy – 4y3 c. 2 38
4x
x
a. The expression is a polynomial with three terms and one variable. The term with the highest degree is 9y2, so the polynomial has degree 2. b. The expression is a polynomial with four terms and two variables. The term with the highest degree is 2x3y2, so the polynomial has degree 5.
c. The expression is not a polynomial because it contains division by the polynomial x + 4.
Slide 20Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Adding like terms
State whether each pair of expressions contains like terms or unlike terms. If they are like terms, then add them.
a. 9x3, −2x3 b. 5mn2, 8m2n
a. The terms have the same variable raised to the same power, so they are like terms and can be combined.
b. The terms have the same variables, but these variables are not raised to the same power. They are therefore unlike terms and cannot be added.
9x3 + (−2x3) = (9 + (−2))x3 = 7x3
Slide 21Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Adding polynomials
Add each pair of polynomials by combining like terms. 2 23 4 8 4 5 3x x x x
2 28 3443 5x x x x
2 2 4 8 34 53x x x x
2 23 4 8 4 5 3x x x x
2 4( ) (3 4 )3) (85x x
2 57x x
Slide 22Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Adding polynomials vertically
Simplify
Write the polynomial in a vertical format and then add each column of like terms.
2 2 2 27 3 7 2 2 .x xy y x xy y
2
2
2
2
7 3 7
2 2
yxy
yx y
x
x
2 25 2 5xyx y
Slide 23Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To subtract two polynomials, we add the first polynomial to the opposite of the second polynomial. To find the opposite of a polynomial, we negate each term.
Slide 24Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Subtracting polynomials
Simplify
The opposite of
3 2 3 25 3 6 5 4 8 .w w w w
3 2 3 25 4 8 is 5 4 8w w w w
3 2 3 25 3 6 5 4 8w w w w
3 2(5 5) (3 4) ( 6 8)w w
3 20 7 2w w 27 2w
Slide 25Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Subtracting polynomials vertically
Simplify
Write the polynomial in a vertical format and then add the first polynomial and the opposite of the second polynomial.
2 210 4 5 4 2 1 .x x x x
2
2
10 4 5
4 2 1
x
x
x
x
26 6 6x x
Slide 26Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Writing and evaluating a monomial
Write the monomial that represents the volume of the box having a square bottom as shown. Find the volume of the box if x = 5 inches and y = 3 inches.
The volume is found by multiplying the length, width, and height together. This can be written as x2y. To calculate the volume let x = 5 and y = 3.
xx
y
x2y = 52 ∙ 3 = 25 ∙ 3 = 75 cubic inches
Slide 27Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Addition and Subtraction of Polynomials
Monomials and Polynomials
Addition of Polynomials
Subtraction of Polynomials
Evaluating Polynomial Expressions
5.2
A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers.Examples of monomials: The degree of monomial is the sum of the exponents of the variables. If the monomial has only one variable, its degree is the exponent of that variable.
Slide 29Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
3 2 9 88, 7 , , 8 , y x x y xy
The number in a monomial is called the coefficient of the monomial.
Slide 30Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Identifying properties of polynomials
Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains and its degree.
a. 9y2 + 7y + 4 b. 7x4 – 2x3y2 + xy – 4y3 c. 2 38
4x
x
a. The expression is a polynomial with three terms and one variable. The term with the highest degree is 9y2, so the polynomial has degree 2. b. The expression is a polynomial with four terms and two variables. The term with the highest degree is 2x3y2, so the polynomial has degree 5.
c. The expression is not a polynomial because it contains division by the polynomial x + 4.
Slide 31Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Adding like terms
State whether each pair of expressions contains like terms or unlike terms. If they are like terms, then add them.
a. 9x3, −2x3 b. 5mn2, 8m2n
a. The terms have the same variable raised to the same power, so they are like terms and can be combined.
b. The terms have the same variables, but these variables are not raised to the same power. They are therefore unlike terms and cannot be added.
9x3 + (−2x3) = (9 + (−2))x3 = 7x3
Slide 32Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Adding polynomials
Add each pair of polynomials by combining like terms. 2 23 4 8 4 5 3x x x x
2 28 3443 5x x x x
2 2 4 8 34 53x x x x
2 23 4 8 4 5 3x x x x
2 4( ) (3 4 )3) (85x x
2 57x x
Slide 33Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Adding polynomials vertically
Simplify
Write the polynomial in a vertical format and then add each column of like terms.
2 2 2 27 3 7 2 2 .x xy y x xy y
2
2
2
2
7 3 7
2 2
yxy
yx y
x
x
2 25 2 5xyx y
Slide 34Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To subtract two polynomials, we add the first polynomial to the opposite of the second polynomial. To find the opposite of a polynomial, we negate each term.
Slide 35Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Subtracting polynomials
Simplify
The opposite of
3 2 3 25 3 6 5 4 8 .w w w w
3 2 3 25 4 8 is 5 4 8w w w w
3 2 3 25 3 6 5 4 8w w w w
3 2(5 5) (3 4) ( 6 8)w w
3 20 7 2w w 27 2w
Slide 36Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Subtracting polynomials vertically
Simplify
Write the polynomial in a vertical format and then add the first polynomial and the opposite of the second polynomial.
2 210 4 5 4 2 1 .x x x x
2
2
10 4 5
4 2 1
x
x
x
x
26 6 6x x
Slide 37Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Writing and evaluating a monomial
Write the monomial that represents the volume of the box having a square bottom as shown. Find the volume of the box if x = 5 inches and y = 3 inches.
The volume is found by multiplying the length, width, and height together. This can be written as x2y. To calculate the volume let x = 5 and y = 3.
xx
y
x2y = 52 ∙ 3 = 25 ∙ 3 = 75 cubic inches
Slide 38Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplication of Polynomials
Multiplying Monomials
Review of the Distributive Properties
Multiplying Monomials and Polynomials
Multiplying Polynomials
5.3
Slide 40Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplying Monomials
A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule for exponents.
Slide 41Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Multiplying monomials
Multiply. a. b.
Solutiona. b.
4 36 3x x 3 4 2(6 )( )xy x y
4 36 3x x 4 3( 6)(3)x
718x
3 4 2(6 )( )xy x y
4 3 26xx y y1 4 3 26x y
5 56x y
Slide 42Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Using distributive properties
Multiply. a. b. c.
a.
3(6 )x 4( 2 )x y (3 5)(7)x
b. 3 36 6( ) 3x x
18 3x
4( ) ( ) ( )( 2 )4 42x y x y
4 8x y
c. 3 5 3( )( ) ( ) ( )757 7x x
21 35x
Slide 43Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Multiplying monomials and polynomials
Multiply. a. b.
Solutiona. b. 24 (3 2)xy x y
23 24 4x yxy xy 212 8xx yy xy
3 3( )ab a b
3 3ab a ab b 4 4a b ab
24 (3 2)xy x y 3 3( )ab a b
3 212 8x y xy
Slide 44Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplying Polynomials
Monomials, binomials, and trinomials are examples of polynomials.
Slide 45Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Multiplying binomials
Multiply ( 2)( 4).x x
2 24 4x x xx
2 2( )( ) ( )( )4 )2 ( )4(x xx x x
2 2 4 8x x x
2 6 8x x
Slide 46Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 47Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Multiplying binomials
Multiply each binomial.a. b.
a.
(3 1)( 4)x x
(3 1)( 4)x x 3 3 4 1 1 4x x x x
23 12 4x x x 23 11 4x x
2( 2)(3 1)x x
2( 2)(3 1)x x b. 2 23 ( 1) 2 3 2 1x x x x 3 23 6 2x x x
Slide 48Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Multiplying polynomials
Multiply each expression. a. b.
a.
24 ( 6 1)x x x
24 4 6 4 1x x x x x
3 24 24 4x x x
2( 2)( 5 2)x x x
b. 2 25 ( 2) 2 2 5 2 2x x x x x x x
3 2 25 2 2 10 4x x x x x
24 ( 6 1)x x x
2( 2)( 5 2)x x x
3 27 8 4x x x
Slide 49Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Multiplying polynomials
Multiply
2 23 ( 3 4 ).ab a ab b
2 233 3 43ab aba ab bab 3 2 2 33 9 12a b a b ab
2 23(3 )4a abab b
Slide 50Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Multiplying polynomials vertically
Multiply
21 (2 3).x x x
22 3
1
x x
x
22 3x x 3 22 3x x x 3 22 4 3x x x
Slide 51Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Special Products
Product of a Sum and Difference
Squaring Binomials
Cubing Binomials
5.4
Slide 53Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 54Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Finding products of sums and differences
Multiply.
a. (x + 4)(x – 4) b. (3t + 4s)(3t – 4s)
a. We can apply the formula for the product of a sum and difference. (x + 4)(x – 4) = (x)2 − (4)2
= x2 − 16
b. (3t + 4s)(3t – 4s) = (3t)2 – (4s)2
= 9t2 – 16s2
Slide 55Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Finding a product
Use the product of a sum and difference to find 31 ∙ 29.
Because 31 = 30 + 1 and 29 = 30 – 1, rewrite and evaluate 31 ∙ 29 as follows.
31 ∙ 29 = (30 + 1)(30 – 1)
= 302 – 12
= 900 – 1
= 899
Slide 56Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 57Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Squaring a binomial
Multiply.
a. (x + 7)2 b. (4 – 3x)2
a. We can apply the formula for squaring a binomial.
(x + 7)2 = (x)2 + 2(x)(7) + (7)2
b.
= x2 + 14x + 49
(4 – 3x)2 = (4)2 − 2(4)(3x) + (3x)2
= 16 − 24x + 9x2
Slide 58Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Cubing a binomial
Multiply (5x – 3)3.
= (5x − 3)(5x − 3)2
= 125x3
(5x – 3)3
= (5x − 3)(25x2 − 30x + 9)
= 125x3 – 225x2 + 135x – 27
– 27 – 150x2 + 45x– 75x2 + 90x
Slide 59Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Calculating interest
If a savings account pays x percent annual interest, where x is expressed as a decimal, then after 2 years a sum of money will grow by a factor of (x + 1)2.
a. Multiply the expression.b. Evaluate the expression for x = 0.12 (or 12%), and
interpret the result.
a. (1 + x)2 = 1 + 2x + x2
b. Let x = 0.12 1 + 2(0.12) + (0.12)2 = 1.2544
The sum of money will increase by a factor of 1.2544. For example if $5000 was deposited in the account, the investment would grow to $6272 after 2 years.
Slide 60Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Integer Exponents and the Quotient Rule
Negative Integers as Exponents
The Quotient Rule
Other Rules for Exponents
Scientific Notation
5.5
Slide 62Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Negative Integers as Exponents
Simplify each expression.a. b. c.
Slide 63Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Evaluating negative exponents
Solution
a.
b.
c.
521
1
8
4( )a b
52
5
1
2
1
2 2 2 2 2
1
32
1
1
818 8
4( )a b 4
1
( )a b
Evaluate the expression.
Slide 64Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using the product rule with negative exponents
Solution
4 28 8
4 28 8 4 ( 2)8 28 64
Simplify the expression. Write the answer using positive exponents. a. b.
Slide 65Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using the rules of exponents
Solution
a.
4 5 6x x x 3 54 3y y
4 5 6x x x 4 ( 5) 6x 5x
b. 3 54 3y y 3 54 3 y y 3 ( 5)12y 212y 2
12
y
Slide 66Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify each expression. Write the answer using positive exponents.a. b. c.
Slide 67Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using the quotient rule
Solution
a.
b.
c.
3
6
10
10
7
3
x
x
2 4
6
24
6
x y
x y
3
6
10
103 610 310
3
1
10
7 3x 4x
2 4
6
24
6
x y
x y
2 4
6
24
6
x y
x y 2 6 4 14x y
1
1000
7
3
x
x
4 34x y3
4
4y
x
Slide 68Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify each expression. Write the answer using positive exponents.
a. b. c.
Slide 69Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Working with quotients and negative exponents
Solution
a.
b.
3
1
3
3 6
5 4
2
6
a b
a b
33 27
c.
3 6
5 4
2
6
a b
a b
4 6
5 3
2
6
b b
a a
10
83
b
a
3
1
3
2
3
3
a
2
3
3
a
23
3
a
6
23
a
6
9
a
Slide 70Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Important Powers of 10
Slide 71Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Number 10-3 10-2 10-1 103 106 109 1012
Value Thousandth Hundredth Tenth Thousand Million Billion Trillion
Write each number in standard form.
a. b.
Slide 72Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Converting scientific notation to standard form
0.0064
Move the decimal point 6 places to the right since the exponent is positive.
3,000,000
Move the decimal point 3 places to the left since the exponent is negative.
63 10 36.4 10
Slide 73Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Write each number in scientific notation.
a. 475,000 b. 0.00000325
475000
Slide 74Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Writing a number in scientific notation
0.00000325
63.25 10
Move the decimal point 5 places to the left.
54.75 10
Move the decimal point 6 places to the right.
Slide 75Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Division of Polynomials
Division by a Monomial
Division by a Polynomial
5.6
Slide 77Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Dividing a polynomial by a monomial
Divide.5 3
2
6 18
6
x x
x
3
2
56 18
6
x x
x
2 2
5 36 8
6 6
1
x x
x x 3 3x x
Slide 78Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Dividing and checking5 4 2
3
16 12 8
4
y y y
y
Check:
5 4 2
3 3 3
16 12 8
4 4 4
y y y
y y y
2 24 3y y
y
3 2 24 4 3y y y
y
3 2 3 3 2
4 4 4 3 4y y y y yy
5 4 216 12 8y y y
Divide the expression and check the result.
5 4 2
3
16 12 8
4
y y y
y