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May 24th

May 24th

Exponents Polynomials

Exponent Review :

Exponential Notation represents
repeated multiplication.

Multiplying with Exponential Values:
If there ARE parenthesis…
Exponent is even = positive answer
(-5)^2 = -5 • -5 = 25
Exponent is odd = negative answer
(-5)^3 = -5 • -5 • -5 = -125

Multiplying with Exponential Values:

Multiplying with exponential values…If
there are NO parenthesis…
Exponent only applies to the value it
touches

Bottom line …a negative value with an exponent and
NO parenthesis will always be a negative answer…

Multiplying with Exponents:

Steps for Multiplying with Exponents:
Step 1: Multiply the numerical coefficients
if necessary
Step 2: Write the base value(s) in the answer
Step 3: Add the exponential value(s) for like
base values

The pattern is…when multiplying with the
values for like (same) base variables.

Watch the pattern…when multiplying same
base variables, add the exponential values
for like base variables…

Multiplying with Exponents Practice:

Power Rules with Exponents:

Steps for Power Rules for Exponents (a Power
Raised to a Power)
Step 1: Perform the exponential operation on
the numerical coefficient if
necessary
Step 2: Write base value(s) in the answer
Step 3: Multiply the exponential values for
the variables with the outside
exponential value then write that

Examples for Power Rules for Exponents:

You don’t work out the number necessarily
unless it is used as a numerical coefficient…

Power Rules with Exponents:

When you have a power raised to a
power, you multiply the exponential
values:

Remember the exponent applies to
everything inside the parenthesis which
includes the numerical coefficient

Power Rules for Exponents Practice:

Division with Exponents ::

Step 1: Divide or reduce the numerical coefficients if
necessary
Step 2: De termine if the base value goes on top, goes on
the bottom, or cancels out…write it in the appropriate spot
(if the larger exponent for the given base is on the
top, that base value goes on top…if the larger
exponent for the given base is on the bottom, that
base value goes on the bottom…if the exponent is
the same on the top and the bottom, the base
value cancels out…)
Step 3: Subtract the exponent values for the
same-based variable(s) and put the answer next to
the appropriate vase value in the answer

Division with Exponents Examples:

Division with Exponents Practice:

Polynomials:

Polynomials are mathematical classifications
for expressions
Ex pressions cannot be classified as a
polynomial if it contains:
- Negative exponent

- Variable in the denominator

Polynomials are always written in
alphabetical order in descending order of
the exponent…

A prime polynomial is a polynomial that is
not factorable…

Classifications of Polynomials:

Monomials – Expressions with 1 term
Examples:
Binomials – Expressions with 2 terms
Examples:

Trinomials – Expressions with 3 terms
Examples:

Polynomials – Expressions with 4 or more terms
Examples:

Classifications of Polynomials Practice:

Classify each polynomial:

 Binomial Trinomial Trinomial Not classified (negative exponent) Binomial Monomial Binomial

Factoring Polynomials:

The different types of factoring involving
polynomials include- Factoring out the Greatest Common
Numerical Factor (GCNF)
- Factoring out the Greatest Common
Monomial Factor (GCMF)
- Factoring into two binomials

Greatest Common Factor (GCNF)

Factor out (or divide out) the largest common
number for all terms.
Step 1: Divide out and write in front of the
parenthesis the largest common numerical
value
Step 2: The remaining values are written
inside the parenthesis

EXAMPLES:

Factoring out the GCNF Practice:

Greatest Common Factor (GCMF)

Factor out (or divide out) the largest common
monomial value(s) for all terms.
Step 1: Divide out and write in front of the
parenthesis the largest common monomial
value (could be number and variable or
just variable)
Step 2: The remaining values are written inside
the parenthesis

Greatest Common Monomial
Factor (GCMF):

EXAMPLES:

Factoring out the GCMF Practice:

Multiplying Polynomials:

Multiplying binomials is one way to use
the distributive property . Once the
multiplying is done, like terms are
combined. There are several ways to
multiply binomials.

The most common method for
multiplying
binomials is called the “FOIL” method.

Each letter in “FOIL” represents the terms that
are multiplied. Use the FOIL method to multiply
binomials

F – first values
O – outside values
I – inside values
L – last values

Multiplying Binomials:

Once you have done all of your multiplying,
combine like terms …if you follow the FOIL
method, the like terms will be the two
values in the middle.

Another way to multiply binomials is to use

 Hint: be sure to put your signs in to know which values are positive and which are negative

This method is the easiest way to multiply
binomials and trinomials

Patterns for Multiplying Binomials

Knowing the patterns that result from
multiplying binomials will make both
multiplying and factoring much easier.

Learn the patterns!

…Notice that when
The terms are identical
(which is called squaring the
binomial
)…
…the first terms, as
always is the variable
squared…
…the middle term is the
two elements of the
original binomial
doubled…
…the last term is a
perfect square…

Special Patterns for Multiplying Binomials:

When the 2nd values are identical…

Squaring a binomial…binomials are identical…

Multiplying Binomials Practice:

Factoring Polynomials into Binomials:

Factoring is working backwards from
“FOILing” or undoing the multiplication of
two binomials.

When you break an equation down into its
basic components, it is called factoring.

Always write polynomials in descending
order of degrees. This is called standard
form.

Methods for Factoring
Trinomials into Two Binomials:

There are many different ways to factor
trinomials into two binomials.

Think back to the patterns that emerged
when multiplying two binomials…

Use the patterns to help with factoring…
Step 1: Identify the pattern and set up
Step 2: Determine the factors that are
used (when multiplies together you
have the third term…when combined
together you have the middle term)
Step 3: Insert factors into the parenthesis

Review the Patterns for Multiplying Binomials:

Special Patterns for Multiplying Binomials:

When the 2nd values are identical…

Squaring a binomial…binomials are identical…

Examples of Factoring a Polynomial:

EXAMPLE 1:

EXAMPLE 2:

EXAMPLE 3:

EXAMPLE 4:

EXAMPLE 5:

EXAMPLE 6:

Factoring Polynomials – Practice:

Factoring Polynomials – Practice:

Now put it all together…factor each…decide
what to factor for…GCNF, GCMF, or as
two binomials…

You MUST practice the
skills in order to get it!!!

You MUST work practice
problems in order to
become proficient at
math!!!

Review the things that you need
to review.

Study the things that you need
to spend more time on.