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Factoring notes and the zero factor property
Factoring is the process of writing a polynomial as the product of two or more simpler
polynomials.
A prime number is a number that is only divisible by one and itself.
A prime polynomial is a polynomial that can’t be factored any further.
Factoring a polynomial
1. The first rule of factoring is to factor out any common factor or factors by factoring by
the greatest common factor method.
2. If the polynomial is a binomial, check to see if it is the difference of two squares, the
difference of two cubes, or the sum of two cubes. The sum of two squares doesn’t
factor.
3. If the polynomial is a trinomial, check to see if it is a perfect square trinomial.
If the polynomial is not a perfect square trinomial, try factoring by the illegal way.
4. If the polynomial has more than three terms try to factor by grouping.
The polynomial may be a more complex difference of two squares or perfect square
trinomial.
5. Factor by substitution problems such as the following example.
(x + 1)2 − 5(x + 1) + 6
If the polynomial is a trinomial with exponents of 4 and 2, we can factor by substitution
but it is not necessary.
x 4 − 7x 2 + 12
Greatest Common Factor
To factor by the greatest common factor method factor out the greatest common
numerical factor common to all terms. Also factor all of the variable(s) by factoring out
the smallest exponent common to all terms. Any answer can be checked by
multiplication.
Factoring by grouping.
Use parentheses and group terms that have common factors. Then factor out what is
common in the first set of parentheses. Next factor out what is common in the second
set of parentheses. You may want to use the distributive property to check your signs.
Next factor out the common binomial term. Be careful when the sign on the third term
is negative. Write the minus sign on the third term as + − . Make sure the minus sign
goes inside the parentheses. Otherwise your signs won’t be correct on the second
binomial term.
Sometimes we are factoring a problem where there is nothing in common except for a 1
or a minus 1 in one or both of the sets of parentheses. Any answer can be checked by
multiplication. I usually group the first and second terms and the third and fourth
terms.
Factor the Greatest Common Binomial Term
Sometimes we factor out the greatest common binomial term, and then factor by
grouping.
(3x + 4 )(x − 2 ) + (3x + 4 )(x + 5 )
(3x + 4 )[ (x − 2) + (x + 5) ]
(3x + 4 )(2x + 3 )
Common binomial term is 3x + 4
First factor out the common binomial term, then
collect like terms inside the bracket.
(4 x + 5 )(x − 2) + (4x + 5)(x + 4 )
(4 x + 5)[ (x − 2) + (x + 4 ) ]
Common binomial term is 4 x + 5
First factor out the common binomial term, then
collect like terms inside the bracket.
Factor out the common factor inside the second
parentheses. Place the common factor in front of
first parentheses.
(4 x + 5 )(2x + 2)
(4 x + 5 )2(x + 1)
2(4 x + 5)(x + 1)
(3x − 2 )(2x + 5) − (3x − 2)(x + 1)
(3x − 2)[ (2 x + 5 ) − (x + 1) ]
(3x − 2)(2 x + 5 − x − 1)
(3x − 2)(x + 4 )
Common binomial term is 3x − 2
First factor out the common binomial term, then
subtract and change the signs on the second
binomial inside the bracket, and collect like terms
inside the bracket.
Factor by the illegal move
We will cover this type of factoring in more notes later on.
Solving an equation by factoring and using the zero-factor property
The zero-factor property is used to solve quadratic equations by factoring.
One side of the equation must be set equal to zero. May first have to do FOIL on one or both
sides of the equation before factoring.
The zero-factor property
If ab = 0 , then either a = 0 or b = 0
For example
(x − 2)(x + 4 ) = 0
Set what is inside each set of parentheses equal to zero and solve for x.
(x − 2)(x + 4 ) = 0
x −2 =0
x =2
x+4=0
x = −4
x (x − 6)(3x + 4) = 0
x =0
x−6 = 0
x =0
x=6
3x + 4 = 0
−4
x=
3
We had three answers in the above examples because of the x on the outside of the first
parentheses.
The following special types of factoring will be covered in more notes later on.
a. Factor perfect square trinomials
b. Factor difference of two squares
c. Factor the sum or difference of two cubes