Polynomials

4-4 to 4-8

Learning Objectives 4-4

Polynomials

Monomials, binomials, and trinomials

Degree of a polynomials

Evaluating polynomials functions

Polynomials

Polynomials are sums of these "variables and exponents"

expressions.

Each piece of the polynomial, each part that is being

added, is called a "term". Polynomial terms have

variables which are raised to whole-number exponents (or

else the terms are just plain numbers).

There are no square roots of variables, no fractional

powers, and no variables in the denominator of any

fractions.

Polynomials

A typical polynomial:

Polynomials

6x –2 This is NOT

a polynomial term...

...because the variable has a

negative exponent.

This is NOT

a polynomial term...

...because the variable is in the

denominator.

This is NOT

a polynomial term...

...because the variable is inside a

radical.

4x2 This IS a polynomial term... ...because it obeys all the rules.

y

x2

x

Polynomial Degrees

Second-degree polynomial, 4x2, x2 – 9, or

ax2 + bx + c

Third-degree polynomial, –6x3 or x3 – 27

Fourth-degree polynomial, x4 or 2x4 – 3x2 + 9

Fifth-degree polynomial, 2x5 or x5 – 4x3 – x + 7

Monomial

An expression containing only one term is called a

monomial.

Example: 7, x, 7x, -6x, ab, etc.

A monomial is a number, a variable, or the product of

a number and one or more variables with whole

number exponents.

Not a monomial: 8 + x, 2/n, 5x, a-1, -3x-3, x2.5

Binomial and Trinomial

Binomial: An expression containing two terms is

called a binomial.

Examples: 7x+5, 6y - p

Trinomial: An expression containing three terms is

called a trinomial.

Examples: 2x+3y-4z

Monomial, Binomial, and Trinomial

Type Definition Example

Monomial A polynomial with one term 5x

Binomial A polynomial with two terms 5x - 10

Trinomial A polynomial with three terms

Evaluating Polynomial Functions

“Evaluating” a polynomial is the same as evaluating

anything else: you plug in the given value of x, and

figure out what y is supposed to be.

Evaluate f(x) = 2x3 – x2 – 4x + 2 at f(-3)

Examples

32)( 3 xxxxh

Find:

a) h(0)

b) h(-3)

Evaluating Polynomial Functions

The revenue ($) that a mfg. of desks receives is given

by the polynomial function: f(d) = -0.08d2 + 100d

where d is the number of desks.

a) Find the total revenue if 625 desks are made.

b) Does increasing the number of desks being made to 650

increase the revenue?

Section 4.4 Review

Polynomials

Monomials, binomials, and trinomials

Degree of a polynomials

Evaluating polynomials functions

Section 4.5 Learning Objective

Adding and subtracting monomials

Adding and subtracting polynomials

Adding and subtracting multiples of polynomials

An application of adding polynomials

Remember Like Terms?

4x and 3 NOT like terms The second term has no variable

4x and 3y NOT like terms

The second term now has a variable,

but it doesn't match the variable of

the first term

4x and 3x2 NOT like termsThe second term now has the same variable,

but the degree is different

4x and 3x LIKE TERMSNow the variables match and the

degrees match

Adding and Subtracting Monomials

Step 1: Remove the ( ).

Step 2: Combine like terms.

Examples:

4ab + (-2ab) =

4ab - (-2ab) =

6x2 - x2 =

Adding Polynomials

Examples:

(5p2 – 3) + (2p2 – 3p3)

(4 + 2n3) + (5n3 + 2)

Subtracting Polynomials

Examples:

(a3 – 2a2) - (3a2 – 4a3)

(4r3 + 3r4) – (r4 – 5r3)

Adding & Subtracting Multiples of

Polynomials

Example:

Add 3(x2 + 4x) and 2(x2 – 4)

Application #1

A house is purchased for $105,000 and is expected to

appreciate $900 per year, its value y after x years is

given by the polynomial function

f(x) = 900x + 105,000.

a) What is the expected values in 10 years?

Application #2

A house second home is purchased for $120,000 and

is expected to appreciate $1,000 per year.

a) Find a polynomial function that will give the

appreciated value y of the house in x years.

b) Find the value of this second house after 12 years.

Section 4.5 Review

Adding and subtracting monomials

Adding and subtracting polynomials

Adding and subtracting multiples of polynomials

An application of adding polynomials

Section 4.6 Learning Objectives

Multiplying monomials

Multiplying a polynomial by a monomial

Multiplying a binomial by a binomial

The FOIL method

Special products

Multiplying a polynomial by a binomial

Multiplying three polynomials

Multiplying binomials to solve equations

Multiplying Monomials

When multiplying two monomials, multiply the

numerical factors and then multiply the variable

factors.

Example:

(5x2y3)(6x3y4)30x33y7

Multiplying a Polynomial by a Monomial

Use the distributive property to remove parentheses

and simplify.

Example:

2x3(3x2 – 5x)

Multiplying a Binomial by a Binomial

Multiply each term of one binomial by each term of

the other binomial and combine like terms.

Example:

(x + 3)(x + 2)

(x + 3y)(2x − 5y)

The FOIL Method

F First terms

O Outside terms

I Inside terms

L Last terms

One way to keep track of your distributive property is to Use the FOIL

method. Note that this method only works on (Binomial)(Binomial).

Find the product of (z + 3)(z + 1)

The Vertical Method and Grid Method

The vertical method:

The grid method:

Multiply: (x + 2)(x + 3)

Examples

Find each product:

1. (3y – 2)2

2. (5t – 2u)(2t + 3u)

Special Products

Square of the sums:

(x + y)2 = X2 + 2xy +y2

The square of the differences:

(x – y )2 = X2 – 2xy +y2

Product of the sum and difference of two terms:

(a + b)(a – b) = a2 – b2

Examples

(z + 6)2

(7x – 2)2

(5m – 9n)(5m + 9n)

Multiplying a polynomial by a binomial

Rule: To multiply one polynomial by another, multiply

each term of one polynomial by each term of the

other polynomial and combine like terms.

Examples

(3x + 2)(2x2 – 4x + 5)

(-2x2 + 3)(2x2 – 4x -1)

Multiplying Three Polynomials

-2y(y + 3)(3y – 2)

Solve the equation:

(x + 2)(x + 3) = x(x + 7)

Dividing by Polynomials Monomials

4

84 b

4

32

10

5

pq

qp

Dividing Polynomials by Polynomials

Divide

Divide

652 2 xxx

1091 2 xxx

Dividing Polynomials by Polynomials

Divide by 5x + 331011 2xx

P339, 22