Name:__________________________________________ Period:___________ Factoring Notes 1

Standard: F.IF.8

Objective: I can factor by pulling out the greatest common factor.

Remember when we factored 1260 or 525

When you are factoring a number you are finding all of the prime numbers that you multiply

together to get the original number. Factoring a polynomial is the same, you are find all the

prime polynomials that you multiply to get the original polynomial. There are a couple of

different methods of factoring.

Factoring by pulling out the greatest common factor.

Steps Example

1. Find the factors that each term has in common. That is the greatest common factor

2. Divide each term by the greatest common factor

3. For the final answer, you write the greatest common factor outside the parenthesis, and inside the parenthesis you write what you get when you divided in step number 2

Examples:

a) b)

Practice A Factor each of the following polynomials. If the polynomial cannot be factored, write prime.

1.

2. 3.

4.

5. 6.

Factoring Notes 2

7.

8. 9.

Standard: F.IF.8

Objective: I can factor by using the difference of squares.

What is the difference of squares?

- When you have a binomial (2 terms) where the two terms are PERFECT SQUARES

and they are also being SUBTRACTED.

To factor a difference of squares

Steps Examples

1. Pull out the greatest common factor, if possible

2. Take the square root of the first term in the parenthesis

3. Take the square root of the second term in the parenthesis

4. Your final answer is in each set of parenthesis. You have the square root of each term, one with a plus sign and one with a subtraction sign

Examples:

a) – b) –

Practice B Factor each polynomial by using the difference of squares. Don’t forget to pull out the greatest common factor first if possible.

1. –

2. – 3. –

4.

5. – 6. –

Factoring Notes 3

7. –

8. – 9. –

Standard: F.IF.8

Objective: I can factor by grouping.

In order to factor by grouping, the polynomial needs to have 4 terms.

Steps Example

1. Pull out the greatest common factor if possible

2. Put the first two terms in a group and the last two terms in a group

3. Looking at only the first group, pull out the greatest common factor

4. Looking at only the second group, pull out the greatest common factor

5. For each group, you should have the same thing left in the parenthesis, pull that out of each term

6. Your final answer is what is left in the parenthesis multiplied by what is outside the parenthesis

Examples:

a) b)

Pactice C Factor each polynomial by grouping.

1.

2.

Factoring Notes 4

3.

4.

5.

6.

7.

8.

9.

10.

Standard: F.IF.8

Objective: I can factor by magic factors.In order to factor by magic factors, you have to

have a trinomial, (3 terms). Every trinomial that is not prime can be factored by using magic

factors. There are other methods for trinomials, but this is the only method that will work on

every trinomial that is not prime. A trinomial is written in the form:

Factoring Notes 5

Steps Example

1. Pull out the greatest common factor if possible

2. Multiply and , then find the all options of two numbers that you can multiply to get that number

3. Of those options, find the one that has

the sum of

4. Rewrite the original polynomial so that it is no longer a trinomial, make it so that is

has 4 terms. To do so, split the term into two terms using the numbers you found in step 3

5. Factor by grouping

Examples:

a) b)

Practice D Factor each polynomial by using magic factors.

1.

2.

3.

4.

Factoring Notes 6

5.

6.

7.

8.

9.

10.

11.

12.

Standard: F.IF.8

Objective: I can factor a perfect square trinomial.

What is a perfect square trinomial?

- A trinomial (3 terms) in which the first and last term are perfect squares and .

A perfect square trinomial can be factored by using magic factors. But there is also a shortcut

method.

Factoring Notes 7

Steps Example

1. Pull out the greatest common factor, if possible

2. Check to make sure it is a perfect square trinomial. (Is the first and last term perfect squares? If they both are,

is the ?)

3. Take the square root of the first term

4. Take the square root of the last term

5. Your final answer is in both parenthesis you have the square root of the first term, the square root of the last term, and the sign between them is the sign

on .

Examples:

a) b)

Practice E Factor each of the following polynomials

1.

2. 3.

4.

5. 6.

7.

8. 9.

10.

11. 12.

Factoring Notes 8

Example:

Example:

Example:

Example:

Example:

Factor by

difference of

squares

Factor by

magic factors

Factor by

grouping

Factor by

perfect

square

trinomial

2 Terms

3 Terms

4 Terms

Pull out

greatest

factor

Factoring Notes 9

Standard F.IF.8

Objective: I can use the best method to decide how to factor a polynomial.

Practice F Factor each polynomial using the best method.

1.

2. –

3.

4.

5.

6.

7. –

8.

Factoring Notes 10

9. –

10.

11.

12. –

13.

14. –

15.

16.

Factoring Notes 11

Standard: F.IF.8

Objective: I can find the zeros of a quadratic function by using the zero product

property.

What is a quadratic function?

- A polynomial in which the degree of the polynomial is 2.

A quadratic function can be written in 3 different forms:

- Standard form:

- Vertex form:

- Factored form:

In all of these cases

What is a Zero?

- The zero of a function is a value of the input that makes the output equal to zero.

The zero of the function are known as roots, x-intercepts, and solutions

When you are finding the zeros of a function, you are substituting 0 in for , the solving for

. But how do you do that since there are two terms with an x in it? One method is to use the

zero product property.

What is the zero product property?

- If the product of two quantities equals zero, at least one of the quantities equals zero. If

, then or .

Using the zero product property to solve a quadratic equation in factored form.

Steps Example

1. Separate each factor into a separate equation that is equal to zero.

2. Solve each equation for x.

Practice H Find the zeros of each of the following quadratic functions.

1.

2. 3.

4.

5. 6.

Factoring Notes 12

Standard F.IF.8

Objective: I can solve a quadratic function by factoring.

Steps Example

1. Factor the polynomial.

2. Use the zero product property to find the zeros.

Example:

a) b) –

Practice I Find the zeros of each of the following functions.

1.

2.

3.

4.

Factoring Notes 13

5. –

6.

7.

8.

9.

10.

Practice J

Find each of the following functions given that , , and

1.

2.

3.

4.

5.

6.

Factoring Notes 14

Simplify each of the following radicals

7.

8. 9.

10.

11.

12.

Simplify each Radical expression

13.

14. 15.

16.

17. 18.

19.

20.

21.