Download Chapter 7: Polynomials

Chapter 7: Polynomials
This chapter starts on page 320, with a list of key words and concepts.
Chapter 7: Get Ready!

Here are the concepts that need to be reviewed before
starting Chapter 7:
1.
Represent expressions using algebra tiles.
2.
The zero principle
3.
Polynomials
4.
Factors
7.1 Add and Subtract Polynomials!
A
term is an expression formed by
the product of numbers and
variables.
 3x2
and 4x are examples of terms.
What is a variable?

A variable is a letter that is used to represent a value
that can change or vary.

For example, in 4x – 1, the variable is x.
The parts of a term

There are 2 parts of a
term:
1.
The numerical
coefficient
2.
The literal
coefficient
Think:
numerical=number
literal=letter
The numerical coefficient
 The
numeric factor of a term is
called the numerical coefficient.
 For
example, the numerical
coefficient of 4x is 4
and for -3y2 it is -3
The literal coefficient
 The
non-numeric factor (i.e. the
letter) of a term is called the literal
coefficient.
 For
example, the literal coefficient
of 4x is x and for -3y2 it is y2.
(You have to include the exponent)
A polynomial
A
polynomial is an algebraic
expression consisting of one or
more terms separated by addition
(+) or subtraction (-) symbols.
Types of polynomials

There are 4 different
types of polynomials:

Monomials

Binomials

Trinomials

Polynomials
The definition of each
polynomial
A
monomial has one term.
A
binomial has two terms.
A
trinomial has three terms.
A
polynomial is the general
expression used when there is more
than one term.
Like terms
 Like
terms are terms that have the
same literal coefficient.
 For
example, 3x and 4x are like
terms because they have the same
literal coefficient, x.
 And
-3y2 and 5y2 are like terms
An algebraic model
 An
algebraic model can represent a
pattern, a relationship or a numeral
sequence.
 An
algebraic model is always
written in the form of an algebraic
expression, algebraic equation or
algebraic formula.
7.3: Multiply a monomial by a polynomial

Here is the distributive property, a
rule that allows you to simplify
expressions involving the
multiplication of a monomial by a
polynomial.

3(x + 2) = 3(x) + 3(2) = 3x + 6
The expansion of
expressions

When you apply the distributive
property, you are expanding an
expression.
7.4: Multiply two
binomials

In order to multiply 2
binomials, we can use:
1.
Area models using
Alge-Tiles.
2.
Multiply using
distributive property
The area of a rectangle
Area of a rectangle = length of
rectangle x width of rectangle
Method #1 (Area models)

When building rectangular tile
models, use these directions:
1. Begin
at the bottom left corner with
x2 tiles first.
2. Construct
a rectangle in the top right
corner with unit tiles.
3. Fill
the top left and bottom right
spaces with x-tiles.
Method #2 Distributive
Property

In order to use this method properly, use these
directions:
1.
Multiply the 2 first terms together in each bracket
2.
Multiply the first term in the first bracket with the
second term in the second bracket
3.
Multiply the second term in the first bracket with the
first term in the second bracket together
4.
Multiply the 2 last terms together
5.
Add all the products together in order to obtain the
simplified expression. (Combine like terms!)
The result of multiplying 2 binomials

When you multiply 2 binomials
together, you will get a trinomial ***

For example:

(x + 2)(x + 3) = x2 + 5x + 6
7.5: Polynomial Division

To divide a polynomial by a monomial, it is like
applying the distributive property in reverse.

For example, (6x + 9) ÷ 3 = (6x/3) + (9/3) = 2x + 3

*** A number divided by itself equals 1. (4÷4=1
and x÷x=1)
7.2: Common Factors

There are 3 ways to
factor a polynomial:
1.
The sharing model
2.
The area model
3.
The greatest common
factor method (GCF)
Factoring a polynomial

In order to factor a polynomial
completely, find the polynomial’s
greatest common factor.

You can find these common factors in
the numerical coefficients, in the
literal coefficients or in the both of
them.
Which method should you
use?

The sharing model works best when
the common factor is a number.

The area model works best when the
common factor is a letter.
An example of factoring

3x + 12 = 3(x + 4)

3x + 12 = 3(x + 4) are equivalent
expressions.
The expanded form

3x + 12 is in the expanded form and
contains two terms.
The factored form

3(x + 4) is in the factored form.

The factored form has 2 types of
factors: 3 is the common numeric
factor and (x + 4) is the polynomial
factor.
7.6: Applying algebraic
modeling

Here is how you can solve an algebraic
word problem:
1.
Read the problem at least 3 times.
2.
Identify the known and unknown
quantities.
3.
Make a plan that will solve for the
unknown quantities.
4.
Solve your problem with the plan that
you came up with in #3.
5.
Write your final answer in a complete
sentence.