Title: Factoring Using the Distributive Property

EQ: How do we factor polynomials by using the distributive property? How

do we solve quadratic equations of the form ax + bx + c?

Warm up: p. 479 #72-86 even

Factoring by using the distributive property

• Express a polynomial as the product of a monomial factor and a polynomial factor.

Example:

Factored form:

Notes cont.• Factoring involves finding the GCFExample:

• Now you write each term as the product of the GCF (divide it out)

Example:

• This will give you the factored form of:

Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF.

Use the Distributive Property to factor .First, find the CGF of 15x and .

Factor each number.

Circle the common prime factors.GFC:

Rewrite each term using the GCF.Simplify remaining factors.Distributive Property

Answer: The completely factored form ofis

Factor each number.

Circle the common prime factors.

Use the Distributive Property to factor.

Rewrite each term using the GCF.

Distributive PropertyAnswer: The factored form of

is

GFC: or

Use the Distributive Property to factor each polynomial.

a.

b.

6

Answer:

Grouping to factor• If a polynomial has four or more terms it helps to

group the polynomial and then factor. This means you take and split the polynomial into pairs.

HINTS for grouping:• There are four or more terms• Terms with common factors should be grouped

together.• The two common factors are identical or additive

inverses of each other.Example:

Factor

Group terms with common factors.

Factor the GCFfrom each grouping.

Answer: Distributive Property

Factor

Answer:

The additive inverse property• Recognizing the polynomial as additive inverses

can be VERY helpful when factoring by grouping. Additive inverses are like (-x-7) (7+x). You know they are additive inverses bc when you add them together the sum is 0. Parenthesis are identical except for signs! You need to pull out a negative in one of the outside numbers

• Example:

Factor

Group terms with common factors.

Parenthesis are identical except for signs! You need to pull out a negative in one of the outside numbers

Answer: Distributive Property

= -3a (-5 + b) + 4 (b – 5)

What is outside goes in one parenthesis and what is inside goes onto another.

Factor

Answer:

Zero product property

• If the product of two factors is 0, then at least one of the two factors is 0.

Solve an equation in factored form

• Set up the two binomials so they are equal to zero and then solve for the variable.

Example:

Solve Then check the solutions.

If , then according to the Zero Product Property either or

Original equation

or Set each factor equal to zero.

Solve each equation.

Answer: The solution set is

Check Substitute 2 and for x in the original equation.

Solve Then check the solutions.

Answer: {3, –2}

Solve and equation by factoring

• Write the equation so it is in the form of ab=0• Then solve for x.

Example:

Solve Then check the solutions.

Write the equation so that it is of the form

Original equation

Solve each equation.

Factor the GCF of 4y andwhich is 4y.

Zero Product Propertyor

Subtract from each side.

Answer: The solution set is Check by substituting

0 and for y in the original equation.

Solve

Answer:

Factoring

• You always need to look for a GCF

• If you are grouping you need to make sure you put like terms together before you put in ( ) That means that you may need to reorganize the polynomial

• Always look for difference of perfect squares