Date: Section:

Objective:

SM 2

Monomial: An expression that is a number, a variable, or numbers and variables multiplied together.

Monomials only have variables with positive whole number exponents and never have variables in the

denominator of a fraction or variables under roots.

Monomials: 2 3 415 , , , 23, ,

8 3

xyzb w x x y− Not Monomials:

1

13 54

1, , , x a z

x

−

Constant: A number by itself with no variables, like 23 or 1.−

Coefficient: The numerical part of a monomial (the number being multiplied by the variables.)

Polynomial: A monomial or several monomials joined by + or – signs.

Terms: The monomials that make up a polynomial. Terms are separated by + or − signs.

Like Terms: Terms whose variables and exponents are exactly the same.

Binomial: A polynomial with two unlike terms. Trinomial: A polynomial with three unlike terms.

Degree of a Monomial: The sum of the exponents on the variables.

Examples: 35x is degree 3 2 43x y is degree 6

7x− is degree 1 (It can also be written as 17x− )

25 is degree 0 (It can also be written as 025x )

Degree of a Polynomial: Find the degree of each term of the polynomial. The degree of the term with

the highest degree is the degree of the polynomial. (If there’s only one variable, it’s the highest exponent

that appears on the variable.)

Example: 6 33 6 5x x x− + The terms have degrees 6, 3, and 1, so the polynomial is degree 6

It isn’t a polynomial if it has: an exponent on a variable that isn’t a positive whole number (negative

exponent, fraction exponent), a variable under a root, a variable in the denominator of a fraction

Examples: Decide whether each expression is a polynomial. If it is, state its degree. If it isn’t, explain

why not.

a) 4 35 2 6x x x+ + b) 54

3a a− − c)

12

2m + d) 26 1c c

−+ −

e) 1

226 5 2z z+ − f) 7 g) 8 3n− − h) 3 2x +

Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms.

• Add or subtract the coefficients. The variables and exponents do not change!

• Remember to subtract everything inside the parentheses after a minus sign. Subtract means

“add the opposite,” so change the minus sign to a plus sign and then change the signs of all the

terms inside the parentheses.

Examples: Simplify each expression.

a) ( ) ( )2 25 2 7 3n n− + − b) ( ) ( )2 22 5 4r r r r+ + −

c) ( ) ( )2 24 3 1 2 5 6x x x x− + + − + − d) ( ) ( )2 27 12 5 6 4 3z z z z+ − + − −

e) ( ) ( )2 22 3 4w w w w+ − + f) ( ) ( )3 2 2 34 2 5u u u u u− + − −

g) ( ) ( )2 26 3 2 4 3x x x x− − + − − − + h) ( ) ( )2 24 12 7 20 5 8y y y y+ − − + −

i) ( ) ( ) ( )2 2 26 5 4 2 3 7m m m m m m+ − − + − j) ( ) ( ) ( )2 22 5 3 4 8k k k k− + + − − − +

Multiplying Polynomials

To multiply two polynomials, multiply each term of one polynomial by each term of the other

polynomial. Then combine any like terms. When you are multiplying two binomials, this is sometimes

called the FOIL Method because you multiply F the first terms, O the outside terms, I the inside terms,

and L the last terms.

Examples: Multiply.

a) ( )27 3 11xy x y xy− + − b) ( ) ( )3 8m m+ −

c) ( )( )3 1 5 2x x+ − d) ( )( )2 5 5 8z z− + − −

e) ( )( )2 4 2 9t t− + f) ( )( )2 22 1 5 4u u− − +

g) ( )2

5z + h) ( )2

2 3x −

i) ( ) ( )3 3n n+ − j) ( )( )5 2 5 2y y− +

k) ( )( )22 3 5 6 7x x x− − + l) ( )( )2 24 7 3 2 8x x x x+ − − +

Date: Section:

Objective:

Rational Function:

Domain of a Rational Function:

�

Examples: Find the domain of the following rational expressions.

a) 2 1

3

x

x

+

− b)

3

7

x +

c) ( )( )

( ) ( )

2 1 4

4 2 1

x x

x x

+ −

+ + d)

( )( )

5 1

4 3 2

x

x x

+

− +

Simplify rational expressions:

1.

2.

3.

Examples: Simplify each rational expression. State its domain. Write prime if it does not simplify.

a) 2 1

2 1

x

x

+

− b)

2

2

x

x

+

+ c)

( ) ( )

2

2 3 1

x

x x

+

+ −

d) ( ) ( )4 1 3

3

x x

x

+ −

− e)

( ) ( )

( )( )

2 7 3 1

3 1 8

x x

x x

− +

+ − f)

( )8 4

8

x +

SM 2

Date: Section:

Objective:

SM 2

Often, it is useful to combine two functions to make a new function. For instance, you may have a function describing the revenue from a product and a function describing the costs of producing the product. By subtracting the two functions, you can create a function describing the profit made from the product. Steps:

1.

2.

3.

Tips:

•

•

•

•

Examples: Let ( ) 3 5f x x= − and ( ) 2 5 2.g x x x= + − Perform the indicated operations.

a) ( ) ( ) ( )h x f x g x= + b) ( ) ( ) ( )h x f x g x= −

c) ( ) ( ) ( )h x g x f x= − d) ( ) ( ) ( )2 3h x f x g x= +

e) ( ) ( ) ( )4h x f x g x= − + f) ( ) ( ) ( )5h x f x f x= −

g) ( ) ( ) ( )h x f x g x= ⋅ h) ( ) ( ) ( )h x f x f x= ⋅

Evaluating Combined Functions 1.

2.

3.

4.

�

Examples: Let ( ) 2 7,f x x= − and let ( ) 2 3.g x x= − + Evaluate the following.

a) ( ) ( )2 1f g+ b) ( ) ( )0 3f g− − c) ( ) ( )2 3 2f g− ⋅

Examples: Let ( ) 3 5f x x= − and ( ) ( )( )3 1g x x x= + − Perform the indicated operations and state the domain

of the new function.

a) ( )( )

( )

g xr x

f x= b) ( )

( )

( )

f xr x

g x=

Domain: Domain:

c) ( )( )

( )

2 f xr x

f x= d) ( )

( )

( )3

g xr x

g x=

−

Domain: Domain:

Examples: Let ( ) 3 5f x x= − and ( ) ( )( )3 1g x x x= + − Evaluate the following functions with the given values

and functions.

a) ( )

( )

2

2

f

g − b)

( )

( )

2 5

1

f

g

−

−

Story Problems Involving Combined Functions

a) A company estimates that its cost and revenue can be modeled by the functions ( ) 20.6 49 150C x x x= + +

and ( ) 100 75,R x x= + where x is the number of items produced. The company’s profit, ,P can be modeled

by ( ) ( ) ( ).P x R x C x= − Find the profit equation and determine the profit when 60 items are produced.

b) A service committee is organizing a fundraising dinner. The cost of renting a facility is $250 plus $3 per

person, or ( ) 3 250,C x x= + where x represents the number of people attending the fundraiser. The

committee wants to charge attendees $20 each or ( ) 20 .R x x= How many people must attend the fundraiser

for the event to raise $500?

Date: Section:

Objective:

History:

For centuries, mathematicians kept running into problems that required them to take the square roots of

negative numbers in the process of finding a solution. None of the numbers that mathematicians were

used to dealing with (the “real” numbers) could be multiplied by themselves to give a negative. These

square roots of negative numbers were a new type of number. The French mathematician René

Descartes named these numbers “imaginary” numbers in 1637. Unfortunately, the name “imaginary”

makes it sound like imaginary numbers don’t exist. They do exist, but they seem strange to us because

most of us don’t use them in day-to-day life, so we have a hard time visualizing what they mean.

However, imaginary numbers are extremely useful (especially in electrical engineering) and make many

of the technologies we use today (radio, electrical circuits) possible.

The number i:

Examples: Express in terms of .i

a) 64− b) 12− c) 49− − d) 18− −

Imaginary Number:

Complex Number:

SM 2

Steps for Adding and Subtracting Complex Numbers:

1.

2.

Examples: Add or subtract and simplify.

a) ( ) ( )2 5 1 3i i+ + − b) ( ) ( )4 3 2 5i i− − − +

c) ( ) ( )3 7 6i− − − − d) ( )5 1i i− −

Steps for Multiplying Complex Numbers

1.

2.

3.

� Why is �� = −�?

Examples: Multiply and simplify. If the answer is imaginary, write it in the form .a bi+

a) 5i i⋅ b) 2 7i i− ⋅ c) 3 5i i− ⋅

d) 9 4− ⋅ − e) 3 5− ⋅ − f) 8 27− − ⋅ −

g) ( )3 2i i− h) ( ) ( )7 3 9 8i i+ − i) ( )( )1 5 6i i− + − −

j) ( )2

2 i− k) ( )( )3 4 3 4i i− + l) ( )2

4 5i− +