Chapter 5 - Exponents and Polynomials
Section 5.1: Adding and Subtracting Polynomials
Objectives:
1. Review combining like terms.
2. Know the vocabulary for polynomials.
3. Evaluate polynomials.
4. Add polynomials.
5. Subtract polynomials.
6. Add and subtract polynomials with more than one variable.
Review Combining Like Terms
Example: Simplify each expression by adding like terms.
Knowing the Vocabulary for Polynomials
A monomial is a term that has no variable in its denominator, and its variables have only whole
number exponents. Thus, a monomial does not have variables with negative exponents in the
numerator, positive exponents in the denominator, or fractional exponents (roots).
For example:
2x3 ,
2 x 3 ,
4
xy ,  3 , 6.8 x , a 3b 4
5
1
2
4x
,  3x ,
5y
3
are monomials.
a 2b
are not monomials.
The degree of a monomial is the sum of the exponents of its variables. A nonzero constant is
considered to be a monomial of degree 0 since, for example,
be a monomial of no degree.
3  3  1  3  x 0 . Zero is considered to
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Any monomial or sum of monomials is called a polynomial. The degree of a polynomial is the largest of
the degrees of its terms (after like terms have been combined).
A polynomial with one term is called a monomial.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.
If a polynomial is of degree 0 or 1, it is called a linear polynomial. If a polynomial is of degree 2, it is
called a quadratic polynomial. If a polynomial is of degree 3, it is called a cubic polynomial.
The leading coefficient of a polynomial in x is the numerical coefficient of the highest power of x.
Examples : Simplify, classify the polynomial if it has a specific name, identify the degree and the leading
coefficient.
1.
4x2  9x  x2  4x  1 
2.
Example: Evaluate Polynomials
1. Find the value of 5h4 – 3h2 + 4h + 7 when h = – 3.
2. Find the value of 5h4 – 3h2 + 4h + 7 when h = 2.
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CAUTION
Use parentheses around the numbers that are substituted for the variable, particularly when
substituting a negative number for a variable that is raised to a power. Otherwise, a sign error may
result.
Adding Polynomials: To add two polynomials, add like terms.
(a) Add 8y 3  7 y 2  y  3 and 6y 3  2 y 2  4 y  1.
(b) Add 7n3  2n and n2  9n  12.
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Subtracting Polynomials: To subtract two polynomials, change all the signs of the
second polynomial and add the result to the first polynomial
(a) Perform the subtraction
3x  5    6 x  4 .
(b) Subtract y3  4 y 2  2 from 7 y 3  8.
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Section 5.2: The Product Rule and Power Rules for Exponents
Objectives:
1. Use exponents.
2. Use the product rule for exponents.
3. Use the rule (am)n = amn.
4. Use the rule (ab)m = ambm.
5. Use the rule (a/b)m = am/bm.
6. Use combinations of the rules for exponents.
7. Use the rules for exponents in a geometry application.
Example: Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate.
Solution: Since 2 occurs as a factor 6 times, the base is 2 and the exponent is 6.
The exponential expression is 26, read “2 to the sixth power” or simply “2 to the sixth.”
Example: Evaluate and give the name the base and exponent of the following.
In summary,
and
are not necessarily the same.
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Properties of Exponents
For nonzero real numbers a and b and integers m and n ,
1.
2.
3.
a a1
( a is any real number.)
a 0 1
a m  a n  a mn
am
 a mn
n
a
4.
n
5.
a 
6.
a n 
1
an
7.
 a  b n  a n  b n
m
 a mn
1
and
a n
an
n
8.
9.
 a 
an

  n
b
 b 
a
 
b
n
b
 
a
n
Examples:
2 2  2 2 
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Using the Rules for Exponents in a Geometry Application
Example: Find a polynomial that represents the area of the geometric figure.
Example: Find a polynomial that represents the area of the geometric figure.
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Section 5.3: Multiplying Polynomials
Objectives:
1. Multiply a monomial and a polynomial.
2. Multiply two polynomials.
3. Multiply binomials by the FOIL method.
(a)
(b)
5x ( 6x + 7 )
– 2h
( – 3h + 8h – 1 )
Multiplying Polynomials: To multiply two polynomials, multiply each term of the second polynomial by
each term of the first polynomial and add the products.
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Multiply
2 y
2
 5  2 y 3  7 y  4  .
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Multiplying Binomials by the FOIL Method
Step 1 Multiply the two First terms of the binomials to get the first term of the answer.
Step 2 Find the Outer product and the Inner product and add them (when possible) to get the
middle term of the answer.
Step 3 Multiply the two Last terms of the binomials to get the last term of the answer.
Use the FOIL method to find
Find the product
 y  4  3 y  2  .
 6a  3b  4a  2b .
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Section 5.4- Special Products
Objectives
1. Square binomials.
2. Find the product of the sum and difference of two terms.
3. Find greater powers of binomials.
Square of a Binomial
The square of a binomial is a trinomial consisting of the square of the first term, plus twice the product
of the two terms, plus the square of the last term of the binomial. For a and b,
(a + b)2 = a2 + 2ab + b2.
Also, (a – b)2 = a2 – 2ab + b2.
Square each binomial.
CAUTION
A common error when squaring a binomial is to forget the middle term of the product. In general,
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Product of the Sum and Difference of Two Terms
(a + b)(a – b) = a2 – b2
Note: The expressions a + b and a – b, the sum and difference of the same two terms, are called
conjugates. In the example above, x + 2 and x – 2 are conjugates.
Finding the Product of the Sum and Difference of Two Terms
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Section 5.5- Integer Exponents and the Quotient Rule
Objectives:
1. Use 0 as an exponent.
2. Use negative numbers as exponents.
3. Use the quotient rule for exponents.
4. Use combinations of rules.
Zero Exponent
For any nonzero real number a, a 0 = 1.
Example: 170 = 1
Negative Exponents
For any nonzero real number a and any integer n, a  n 
1
1
an
and
n
a n
a
Example: Simplify by writing with positive exponents. Assume that all variables represent nonzero real
numbers.
Changing from Negative to Positive Exponents
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For any nonzero numbers a and b and any integers m and n,
n
 a 
an

  n
b
 b 
Example:
35 24

24 35
and
3
a
 
b
n
b
 
a
n
3
4
5
and      .
5
4
Quotient Rule for Exponents
For any nonzero number a and any integers m and n,
am
mn

a
an
Example:
58
 586 =52 =25.
6
5
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x 10  x 7  x 
a 10

6
a
 5x 7 
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(x 3 ) 7 
y 2 y 7

3
y
3
  3m 2 
 3  
 n

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Section 5.6 -Dividing a Polynomial by a Monomial
Example: Divide
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Example: Divide
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Section 5.8- An Application of Exponents: Scientific Notation
Objectives:
1.
Express numbers in scientific notation.
2.
Convert numbers in scientific notation to numbers without exponents.
3.
Use scientific notation in calculations.
Expressing Numbers in Scientific Notation
Numbers occurring in science are often extremely large or extremely small. Because of the difficulty of
working with many zeros, scientists often express such numbers with exponents, using a form called
scientific notation.
In scientific notation, there is always one nonzero digit before the decimal point.
Note
In scientific notation, the times symbol, x, is commonly used.
Writing a Number in Scientific Notation
Step 1 Move the decimal point to the right of the first nonzero digit.
Step 2 Count the number of places you moved the decimal point.
Step 3 The number of places in Step 2 is the absolute value of the exponent on 10.
Step 4 The exponent of 10 is positive if the original number is greater than the number in Step 1; the
exponent is negative if the original number is less than the number in Step 1. If the decimal point is not
moved, the exponent is 0.
Example 1: Write each number in scientific notation.
a. 153,000,000,000
b. 9547.3
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c. 0.00000005842
d.
e.
f.
g.
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