Download Intermediate Algebra: A Graphing Approach, 4ed

§ 5.2
More Work with
Exponents and
Scientific Notation
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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The Power Rule
The Power Rule and Power of a Product or Quotient
Rule for Exponents
If a and b are real numbers and m and n are integers, then
(am)n = amn
Power Rule
(ab)n = an · bn Power of a Product

a
b
n
an
 n
b
Power of a Quotient
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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The Power Rule
Example:
Simplify each of the following expressions.
(23)3 = 23·3 = 29 = 512
(x4)2 = x4·2 = x8
(5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3
4
 
 
 p 
p
 3  
3
3
r
3r


2
2 4
4

p

3 r 
2 4
4
3 4
p8
 12
81r
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Summary of Exponent Rules
If m and n are integers and a and b are real numbers, then:
Product Rule for exponents am · an = am+n
Power Rule for exponents (am)n = amn
Power of a Product (ab)n = an · bn
n
an
a
Power of a Quotient    n , b  0
b
b
Quotient Rule for exponents
am
mn

a
, a0
n
a
Zero exponent a0 = 1, a  0
Negative exponent a  n 
1
, a0
n
a
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Simplifying Expressions
Simplify by writing the following expression with
positive exponents or calculating.
 3 ab 
  4 7 3 
3 a b 
2
3
2

3
3
2
4
3
ab
7
ab


3  a  b 

 3  a  b 
2
3  2
Power of a quotient rule
4
6
2
 2 2
4 2
3 2
7 2
2
3  2
Power of a product rule
8
3a b
a
6( 14) 26
48
4
8 8
 8 14 6  3 a
b 3 a b  4 8
3a b
3b
Product rule
a8

81b 4
Power rule
Quotient rule
Negative exponents
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Operations with Scientific Notation
Multiplying and dividing with numbers written in scientific
notation involves using properties of exponents.
Example
Perform the following operations.
1)
(7.3  102)(8.1  105) = (7.3 · 8.1)  (102 · 105)
= 59.13  103
= 59,130
1.2 10 4 1.2 10 4
5

0
.
3

10


 0.000003
2)
9
9
4 10
4 10
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Operations with Scientific Notation
Multiplying and dividing with numbers written in
scientific notation involves using properties of exponents.
Example
Perform the following operations.
1) (7.3  102)(8.1  105) = (7.3 · 8.1)  (10-2 ·105)
= 59.13  103
= 59,130
1.2 10 4 1.2 10 4
5

0
.
3

10


 0.000003
2)
9
9
4 10
4 10
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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§ 5.3
Polynomials and
Polynomial Functions
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Polynomial Vocabulary
Term – a number or a product of a number and
variables raised to powers
Coefficient – numerical factor of a term
Constant – term which is only a number
Polynomial is a sum of terms involving
variables raised to a whole number exponent,
with no variables appearing in any
denominator.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Polynomial Vocabulary
In the polynomial 7x5 + x2y2 – 4xy + 7
There are 4 terms: 7x5, x2y2, -4xy and 7.
The coefficient of term 7x5 is 7,
of term x2y2 is 1,
of term –4xy is –4 and
of term 7 is 7.
7 is a constant term.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Types of Polynomials
Monomial is a polynomial with one term.
Binomial is a polynomial with two terms.
Trinomial is a polynomial with three terms.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Degrees
Degree of a term
To find the degree, take the sum of the exponents
on the variables contained in the term.
Degree of a constant is 0.
Degree of the term 5a4b3c is 8 (remember that c
can be written as c1).
Degree of a polynomial
To find the degree, take the largest degree of any
term of the polynomial.
Degree of 9x3 – 4x2 + 7 is 3.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Combining Like Terms
Like terms are terms that contain exactly the same variables
raised to exactly the same powers.
Warning!
Only like terms can be combined through addition and
subtraction.
Example:
Combine like terms to simplify.
x2y + xy – y + 10x2y – 2y + xy
= x2y + 10x2y + xy + xy – y – 2y
(Like terms are grouped together)
= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Adding Polynomials
Adding Polynomials
To add polynomials, combine all the like terms.
Example:
Add.
(3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3
= 4x2 + 3x – 3x – 8 + 3
= 4x2 – 5
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Subtracting Polynomials
Subtracting Polynomials
To subtract polynomials, add its opposite.
Example:
Subtract.
4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8
(– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)
= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7
= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Adding and Subtracting Polynomials
In the previous examples, after discarding the
parentheses, we would rearrange the terms so
that like terms were next to each other in the
expression.
You can also use a vertical format in
arranging your problem, so that like terms are
aligned with each other vertically.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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§ 5.4
Multiplying Polynomials
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Multiplying Two Polynomials
To multiply any two polynomials, use the
distributive property and multiply each term of
one polynomial by each term of the other
polynomial. Then combine like terms.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Example:
Multiply each of the following.
1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3
2) (4x2)(3x2 – 2x + 5)
= (4x2)(3x2) – (4x2)(2x) + (4x2)(5)
= 12x4 – 8x3 + 20x2
(Distributive property)
(Multiply the monomials)
3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5)
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Example:
Multiply (3x + 4)2
Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4).
(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4)
=
9x2 + 12x + 12x + 16
=
9x2 + 24x + 16
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Example:
Multiply (a + 2)(a3 – 3a2 + 7).
(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7)
a4 – 3a3 + 7a + 2a3 – 6a2 +
14
= a4 – a3 – 6a2 + 7a + 14
=
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Example:
Multiply (3x – 7y)(7x + 2y)
(3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y)
= 21x2 + 6xy – 49xy + 14y2
= 21x2 – 43xy + 14y2
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Example:
Multiply (5x – 2z)2
(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z)
= 25x2 – 10xz – 10xz + 4z2
= 25x2 – 20xz + 4z2
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
Example:
Multiply (2x2 + x – 1)(x2 + 3x + 4)
(2x2 + x – 1)(x2 + 3x + 4)
= (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4)
=
2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4
=
2x4 + 7x3 + 10x2 + x – 4
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Multiplying Polynomials
You can also use a vertical format in arranging
the polynomials to be multiplied.
In this case, as each term of one polynomial is
multiplied by a term of the other polynomial,
the partial products are aligned so that like
terms are together.
This can make it easier to find and combine like
terms.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Example:
Multiply (2x – 4)(7x + 5)
(2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5)
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
We multiplied these same two binomials together in the
previous section, using a different technique, but arrived at the
same product.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Special Products
Square of a Binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Product of the Sum and Difference of Two Terms
(a + b)(a – b) = a2 – b2
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Special Products
Although you will arrive at the same results
for the special products by using the
techniques of this section or last section,
memorizing these products can save you some
time in multiplying polynomials.
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Evaluating Polynomials
We can use function notation to represent polynomials.
For example, P(x) = 2x3 – 3x + 4.
Evaluating a polynomial for a particular value involves
replacing the value for the variable(s) involved.
Example:
Find the value P(2) = 2x3 – 3x + 4.
P(2) = 2(2)3 – 3(2) + 4
= 2(8) + 6 + 4
= 6
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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Evaluating Polynomials
Techniques of multiplying polynomials are often useful
when evaluating polynomial functions at polynomial values.
Example:
If f(x) = 2x2 + 3x – 4, find f(a + 3).
We replace the variable x with a + 3 in the polynomial function.
f(a + 3) = 2(a + 3)2 + 3(a + 3) – 4
= 2(a2 + 6a + 9) + 3a + 9 – 4
= 2a2 + 12a + 18 + 3a + 9 – 4
= 2a2 + 15a + 23
Martin-Gay, Intermediate Algebra: A Graphing Approach, 4ed
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