Unit 6 – Exponents, Polynomials, and Polynomial Functions
Basic idea of exponents:
2 5 2 2 2 2 2 32
What is
3  2
?
    2 5
Product Rule for Exponents
If m and n are any real numbers and a is any real number, then a m  a n  a
Examples:
 7  3  3 11
5 5 5 5 1  5

7 p q

2 p q

2 4
 5 4
7 2 3   1  2  14
since the bases aren’t the same the Product Rule does not apply.
Zero exponent
Use the product rule to solve
2  0
 0  4  4 2
so 4
0 must equal 1. If a is any real number, then a 0  expression 0
0
is not defined.
1
. Note that the
Examples:
6 0  1
  0
 1
0
 1
(the last assumes k is not 0)
Negative exponents
Use the product rule to solve
2   2  8 2  
 8 0  1
8 8  2
What is a reciprocal? (a number that when multiplied by another number results in 1)
How many reciprocals can a number have? (1)
So this must mean that 8
But
1
8 2
-2 is the reciprocal of 8
is the reciprocal of 8 2 .
2 .
For any natural number n and any nonzero real number a a  n  a
1 n
Examples:
2  3 
1
2 3
  simplify
3  1  4  1
 3

5
1
  3
5 x  3  5
 1 
 x 3 
4 3 7
3 4 12 12 12
 x
5
3
Caution: remember that you cannot have values that would give you a zero in the denominator!
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Quotient Rule for Exponents a a a a
5
2
2
5


   
   
 a
1
3
 a  3
3
If a is any nonzero real number and m and n are integers, then a a n m
Examples:
3 7
 3 
3 2 a 3 b 4
3 5 x x
 5
 8
 x  x 3
rule doesn’t apply because bases are different
 a
Power Rules for Exponents
  n
 a mn
to raise a power to a power, multiply exponents
  m
to raise a product to a power, raise each factor to that power
  b m
to raise a quotient to a power, raise the numerator and denominator to that power
Examples:
  3
 x   x 24
2
3
4 2 4 16
3 4 81

6 x 7
 2
 6 2 x   36 x 14
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Summary of Exponent Rules
For all integers m and n and all real numbers a and b , the following rules apply:
Product Rule
Quotient Rule
Zero Exponent
Negative Exponent a a m m
 a
 n a
 a a a
0 n a  n


1
1 a n
(a ≠ 0)
  n
 a mn
Power Rules
  m
  b m
(b ≠ 0) a
1
 n
 a n
(a ≠ 0)
Special Rules n
 n

1
(a ≠ 0)


 a
 n b
 m




 n b m a n




(a, b ≠ 0)


 n
(a, b ≠ 0)
Scientific Notation
A number is written in scientific notation when it is expressed in the form a

10 n
1
 a

10 and n is an integer.
where
Converting to scientific notation
1.
Position the decimal point – Place a caret to the right of the first nonzero digit. This is where the decimal point will go.
2.
Determine the numeral for the exponent – Count the number of digits from the existing decimal point to the caret. This number gives the absolute value of the exponent on 10.
3.
Determine the sign for the exponent – Decide whether multiplying by 10 n should make the result from Step 1 larger or smaller. The exponent should be positive to make the number larger and negative to make it smaller.
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Converting from scientific notation

Multiplying a number by a positive power of 10 makes the number larger, so move the decimal point to the right n places if n is positive in 10 n .

Multiplying a number by a negative power of 10 makes the number smaller, so move the decimal point to the left n places if n is negative in 10 n

If n is 0, leave the decimal point where it is.
.
Polynomial – a term or a finite sum of terms in which all variables have whole number exponents and no variables appear in denominators or under radicals.
Polynomials: 3 x

5 4 m
5 
3 m
2 p

8

3 x
3 y
5 z
2
Not Polynomials: x

1 y

4
5
 y
3 x
Typically write a polynomial in descending powers of x: x
5 
6 x
2 
12 x

5
Special names for polynomials based on the number of terms.
Monomial – one term
Binomial – two terms
Trinomial – three terms
The degree of a term in with one variable is the exponent on the variable. The degree of a term with more than one variable is the sum of the exponents on the variables. The degree of a polynomial with multiple terms is the greatest degree of any term.
The degree of 2 x
4
is 4. The degree of
2
5 y
6 x is 8.
The degree of 4 x
3 
2 x
2 
6 x

8 is 3.
Combining like terms
Remember, we simplify polynomials by combining like terms. x
3 
4 x
2 
5 x
2 
1 = x
3 
9 x
2 
1
Remember that like terms have to have the same variables to the same powers!

To add polynomials, combine like terms.
4 x
3 
5 x
2 
3 x

2 y
2 
3 y
 
2 x
3  x
2 
5 y
2
6 x
3 
6 x
2 
3 x

7 y
2 
3 y

6

6

If it helps, when adding polynomials place them vertically and align the like terms to make the process a little clearer.
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Subtracting polynomials follows the same process with one change. To subtract two polynomials, add the first polynomial and the negative of the second polynomial. (The negative of a polynomial is defined as that polynomial with the sign of every coefficient changed.)
Example: Subtract


6 x
2 
8 x

5
 

5 x
2 
7 x

8

Change every sign in the second polynomial and add. (What you are actually doing is distributing the “-1” outside the parentheses to end up with

6 x 2 
8 x

5

5 x 2 
7 x

8
  x 2 
15 x

13
We looked at linear functions in Unit 4. These are ‘first degree polynomials’ and are in the form f
 
 mx
 b . (Note the exponent on the variable is 1, thus this is a first degree polynomial.)
A more general definition of polynomial functions is:
A polynomial function of degree n is defined by f
 
 a n x n  a n

1 x n

1     a
1 x
 a
0
where a n
≠
0 and n is a whole number.
Example: f
 

5 x
2 
3 x

2
Evaluate a polynomial function by substituting the given value for the independent variable and solving. In the above example function, f
   
2 
3
 

2

5

4

6

2

24
We can add and subtract polynomial functions. It looks different but is exactly the same process as adding polynomials.
If f(x) and g(x) define functions then:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) - g(x) .
In each case the domain of the function is the intersection of the domains of f(x) and g(x ).
Example: for the polynomial functions defined as f
 
 x
2 
3 x

7 and g
 
3 x
2 
7 x

7
The sum:

(f + g)(x) = f(x) + g(x) x
2 
3 x

7
 

3 x
2 
7 x

7

 
2 x
2 
10 x

14
The difference:

(f – g)(x) = f(x) - g(x) x
2 
3 x

7
 

3 x
2 
7 x

7


4 x
2 
4 x
Graphing basic polynomial functions
Use your graphing calculator to practice graphing functions and “shifting” the graphs of those functions
Graph
Graph f
 
 x f
 
 x
2
(the identity function)
(the squaring function)
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Graph f
 
 x
3
(the cubing function)
Graph
Graph f f
 
 
 x
2
 x
2

5

5
Graph
Graph f f
   x

5

2
   x

5

2
Graph the squaring function and shift it left 3 units and up 2 units.
Graph the cubing function and shift it right 4 units and down 4 units.
Graph the squaring function, but make it open downward instead of upward.
Graph the squaring function but make it “wider” by a factor of 10.
Graph the squaring function but make it “narrower” by a factor of 5.
Multiplying polynomials is all about using the properties of numbers, such as distributive, commutative and associative and the rules of exponents.
Examples:
Multiplying monomials:

4 a
3
 
 
4

3
 a
3  a
5  
12
 a
3

5  
12 a
8
Multiplying polynomials:
2

8 x
3 
9 x
2
   

9 x
2


16 x
3 
18 x
2


 x

 x x
1


1 x x

 x
3 x


  
1 x

1

  x

3

3
distributive property
 
distributive property again

 x
2 x
2

 x
2 x



3 x
3
 
3
2

 x
2
2
2 x x

2
2 x



 x x
1

 x x
1
 x
3


  

1 x x


1 x
 

3

distributive property
 

distributive property again




2 x
2
 x
2  x
 
3 x

 
3

2

2
2 x x x
2

2
2 x

2
2 x
2

 
2

 x
4

2 x x
3
    
2

1

combine like terms


2 x


6
 x
2
 

 
distributive property yet again
rules of exponents

2 x
4 
4 x
3 
6 x
2
Multiplying binomials
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Using the distributive property when multiplying binomials gives rise to the special method called FOIL. To use FOIL when multiplying binomials:
Multiply the First terms in each binomial
Multiply the Outside terms of each binomial
Multiply the Inside terms of each binomial
Multiply the Last terms of each binomial
And then combine like terms.
Special types of binomials
Product of the sum and difference of the same two terms (i.e. x and y).
 x
 y
 x
 y

FOIL gives us x
2  xy
 xy
 y
2
Combining like terms results in x
2  y
2
Leading to the general rule that the product of the sum and difference of two terms x and y is the difference of the squares of the terms.
Product of the square of a binomial
 x
 y

2
Multiply out to get
 x
 y
 x
 y

FOIL results in x
2  xy
 xy
 y
2
Combining like terms results in
 x
 y

2
Multiply out to get
 x
 y
 x
 y
 x
2
Foil results in x
2  xy
 xy
 y
2

2 xy
 y
2
Combining like terms results in x
2 
2 xy
 y
2
Leading to the general rule that the square of a binomial is the sum of the square of the first term, twice the product of the terms and the square of the last term.

Using binomial methods on more complicated polynomials

3 x

2


5 y
  
3 x

2


5 y

the trick is to pretend that (3x-2) is a single term and then this becomes the product of the sum and difference of two terms or the difference of the squares of
 the terms.
3 x

2
  
2

9 x
2 
12 x

4


now you have the square of a binomial and the square of a monomial
25 y
2
or just 9 x
2 
12 x

4

25 y
2
Multiplying polynomial functions
If f(x) and g(x) define functions, then (fg)(x) = f(x) • g(x)
. The domain of the product function is the intersection of the domains of f(x) and g(x) .
All the same rules and methods for multiplying polynomials apply to multiplying polynomial functions.
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Unit 6 – Exponents, Polynomials, and Polynomial Functions
Dividing by a monomial
To divide a polynomial by a monomial, divide each term in the polynomial by the monomial, and then write each quotient in lowest terms.
Examples:
15 x
2 
12 x

6
3
15 x
2
5 x
3
2 

12 x
4 x
3


6
3
divide each term by 3
2 write each quotient in lowest terms
5 x
3 
9 x
2 
10 x
5 x
2
5 x
3
5 x
2
5
5

9 x
2
2

10
2 x
divide each term by 5x
2
5 x 5 x
 

9
5
 

10
5
 
commutative property and rules of exponents x

9
5

  1 x


simplify and rules of exponents x

9
5

2 x
final simplification (note that this answer is not a polynomial. The quotient of two polynomials is not necessarily a polynomial.)
Divide a polynomial by a polynomial of two or more terms
This process is similar to dividing whole numbers.
Examples:
2 x
2 x
 x

2

10
Write as a division problem, making sure both polynomials are in descending powers of exponents. x

2 2 x
2  x

10
Divide the first term of 2 x 2  x

10 by the first term of x

2 x

2 2 x
2  x
2 x

10
Now multiply x

2 by x x

2 2 x
2  x
2 x

10
2x
2 – 4x
Now subtract the polynomials.
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Unit 6 – Exponents, Polynomials, and Polynomial Functions x

2 2 x
2  x
2 x

10
2x 2 – 4x
5x – 10
Now divide the first term of
2 x

5
5 x

10 by the first term of x

2 . x

2 2 x
2  x

10
2x
2 – 4x
5x – 10
Now multiply 5 by x

2 and place the result underneath.
2 x

5 x

2 2 x 2  x

10
2x
2 – 4x
5x – 10
5x – 10
0
At this point you are finished. The remainder is 0. You should check your answer by multiplying 2 x

5 times x

2 . Your answer should be 2 x
2  x

10 .
If one of the polynomials has a missing term, put it in with a 0 as the coefficient. In other words, if one of the polynomials is 3 x
3 
2 x

5 you would want to insert 0 x
2
into it (i.e.
3 x
3 
0 x
2 
2 x

5 ).
You keep dividing until you have either a remainder of zero or a remainder of smaller degree than the divisor. If you do have a remainder, write remainder divisor
as part of the quotient.
Be sure the terms of each polynomial are written in descending order.
Write any missing terms with 0 placeholders.
Dividing Functions
If f(x) and g(x) define functions, then

 f g


 
 f g
 
  . The domain of the quotient function is the intersection of the domains of f(x) and g(x) , excluding any values of x for which g(x) = 0.
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