21 terms

Monomial

(Also known as a term)

An expression that is one of the following

a number

a variable

the product of numbers and/or variables

Ex: -abc

An expression that is one of the following

a number

a variable

the product of numbers and/or variables

Ex: -abc

Polynomial

(Many)

The addition and subtraction of monomials

Example: y^2 + 7y - 9

The addition and subtraction of monomials

Example: y^2 + 7y - 9

Binomial

2 term polynomial

Example: 5x + 7

Example: 5x + 7

Trinomial

3 term polynomial

Example: y^2 + 7y - 9

Example: y^2 + 7y - 9

Degree of a monomial

The sum of the exponents of variables

Examples:

5x degree: 1

6x^3y^2 degree: 5

Examples:

5x degree: 1

6x^3y^2 degree: 5

Degree of a polynomial

The largest exponent

Examples

3x^4 + 5x^2 - 7x + 1

Degree = 4

Examples

3x^4 + 5x^2 - 7x + 1

Degree = 4

Descending Order

Variables should be written in alphabetical order

When there is more than one term with the sane term with the same variable, the term with the largest exponent will appear first

Variable must be in alphabetical order and within the same variable, must be written in descending order

Example: 14c^3 - c^2 + 2c +16

When there is more than one term with the sane term with the same variable, the term with the largest exponent will appear first

Variable must be in alphabetical order and within the same variable, must be written in descending order

Example: 14c^3 - c^2 + 2c +16

Adding Polynomials

CLT = Combining Like Terms

1) Add the coefficients

2) Keep the variable and their exponents exactly the same

Examples:

(17x^3 - 4x^2 - 3x) + (2x + 5x^2 + 4x^3) = 21x^3 + x^2 + x

1) Add the coefficients

2) Keep the variable and their exponents exactly the same

Examples:

(17x^3 - 4x^2 - 3x) + (2x + 5x^2 + 4x^3) = 21x^3 + x^2 + x

Subtracting Polynomials

1) Add the opposite of all terms in the parentheses that appear after the subtraction sign. Note: this is the same thing as distributing the negative to all of the terms that appear after the subtraction sign

2) Follow the adding polynomials rules

Example:

(9y^2 - 3y + 4) - (5y + 6) = 9y^2 -8y - 2

2) Follow the adding polynomials rules

Example:

(9y^2 - 3y + 4) - (5y + 6) = 9y^2 -8y - 2

Multiplying Polynomials

There are two main techniques:

1) FOIL 2) BOX (chart)

1) FOIL 2) BOX (chart)

FOIL

a shortcut for the distributive property and can be used only when you multiply two binomials

F multiply the first terms of each binomial

O outer

I Inner

L last

Ex. (y + 3) (y + 7)

y^2 + 7y + 3y + 21

y^2 + 10y + 21

F multiply the first terms of each binomial

O outer

I Inner

L last

Ex. (y + 3) (y + 7)

y^2 + 7y + 3y + 21

y^2 + 10y + 21

BOX Method

The box method works for all types of polynomial multiplication

Steps for the chart (box) method:

1) What are the dimensions of the chart

2) Draw and label the chart with each term

3) Multiply the terms and fill in the products in the individual boxes

4) Add the boxes of like terms (CLT) to determine the final answer

5) Write answers in descending order

Steps for the chart (box) method:

1) What are the dimensions of the chart

2) Draw and label the chart with each term

3) Multiply the terms and fill in the products in the individual boxes

4) Add the boxes of like terms (CLT) to determine the final answer

5) Write answers in descending order

Dividing Polynomials by Monomials

1) divide each term in the numerator (dividend) by the monomial denominator(divisor)

Note: Assume no denominator/divisor = 0

2) simplify completely

Ex: 12x-9y-3/3

Follow order of operations (divide first then + or -)

When subtracting, subtract the entire answer of the second division problem

Ex: 3x-6/3=4-8x/2

Note: Assume no denominator/divisor = 0

2) simplify completely

Ex: 12x-9y-3/3

Follow order of operations (divide first then + or -)

When subtracting, subtract the entire answer of the second division problem

Ex: 3x-6/3=4-8x/2

Degrees of Monomials and Polynomials

Use Zoom 6

The degree of a monomial is the sum of the exponents of all its variable

The degree of a polynomial is the largest degree of any of its terms after it has been simplified

Ex: 4x^2-11x-5

The degree of a monomial is the sum of the exponents of all its variable

The degree of a polynomial is the largest degree of any of its terms after it has been simplified

Ex: 4x^2-11x-5

Greatest Common Factor: Gotta Check First

Definitions:

Prime number: A # with only 2 factors, 1 and itself.

Composite: A # with 3 or more factors

Note: 1 is NOT prime and 1 is NOT composite because 1 has only 1 factor

Prime number: A # with only 2 factors, 1 and itself.

Composite: A # with 3 or more factors

Note: 1 is NOT prime and 1 is NOT composite because 1 has only 1 factor

Greatest Common Factor

The biggest number that is a factor of the 2 or more terms. It is found by finding the product of the common prime numbers

prime: 2, 3, 5, 7, 9, 11, 13, 17, 19

prime: 2, 3, 5, 7, 9, 11, 13, 17, 19

What is factoring?

Factoring is the process of converting an algebraic expression to a multiplication problem (or a product of factors)

Polynomial ---- Multiplication Problem

It's the reverse of multiplication!

We start with a polynomial and then through the process of factoring, we end up with the product of two (or more) factors.

Factoring is when we break down the polynomial by stripping away common factors....or factors that can divide evenly into two or more values. We can factor numbers and variables

Polynomial ---- Multiplication Problem

It's the reverse of multiplication!

We start with a polynomial and then through the process of factoring, we end up with the product of two (or more) factors.

Factoring is when we break down the polynomial by stripping away common factors....or factors that can divide evenly into two or more values. We can factor numbers and variables

Factoring Trinomials using the X-BOX Method!

ax^2 + or - bx + or - c

Check for the GCF first

Follow steps on sheet

Check for the GCF first

Follow steps on sheet

Factoring Trinomials using the Grouping Method

1. Multiply the numbers found in the quadratic term and the constant term. (a x c)

2. Using the circle, find the Magic Pair

3. Re-write the "middle man" as the sum/diff of the magic pair (how FOIL would be written out(

4. Group the first two terms and the last two terms

5. Factor out the GCF in each pair/set of terms

6. Factor out the common factor/parenthesis and put the left-overs in another set of parenthesis

7. Check by FOIL

2. Using the circle, find the Magic Pair

3. Re-write the "middle man" as the sum/diff of the magic pair (how FOIL would be written out(

4. Group the first two terms and the last two terms

5. Factor out the GCF in each pair/set of terms

6. Factor out the common factor/parenthesis and put the left-overs in another set of parenthesis

7. Check by FOIL

Difference of Squares

Check to see if the binomial is a Difference of Squares

a) Are BOTH terms perfect squares?

b) Is there a subtraction sign between the two terms

If YES to both a and b, than determine the answer

1) Parenthesis-Write 2 sets ( ) ( )

2) Signs-one ( ) will be +,one ( ) will be -

3) 1st term-Take square root of 1st term in the problem

4) 2nd Term-Take square root of 2nd term in the problem

5) Check by FOIL. Multiply your answer. The result should be the original problem.

a) Are BOTH terms perfect squares?

b) Is there a subtraction sign between the two terms

If YES to both a and b, than determine the answer

1) Parenthesis-Write 2 sets ( ) ( )

2) Signs-one ( ) will be +,one ( ) will be -

3) 1st term-Take square root of 1st term in the problem

4) 2nd Term-Take square root of 2nd term in the problem

5) Check by FOIL. Multiply your answer. The result should be the original problem.

Dividing Polynomials by Binomials

Factoring Method

1) Factor the polynomial(s)

2) Easier to write problem in fractional format

3) Wipe out/cancel common factor groups

1) Factor the polynomial(s)

2) Easier to write problem in fractional format

3) Wipe out/cancel common factor groups