Factoring Tutorial
Introduction
Factoring is used to write an expression as 2 more terms multiplied together.
For example, 10 can be written as (5)(2); therefore 5 and 2 are factors of 10.
We can also do this with polynomials. (Remember – what is a polynomial?
Give some examples of polynomials.
Greatest Common Factors
What is a common factor?
A common factor is one number or expression that will divide into several
numbers or expressions.
Example 1:
25
35
What number divides into both 25 and 35?
Example 2:
14𝑦 5 − 4𝑦 3 + 2𝑦
What monomial will divide into all 3 terms?
Divide each term by the GCF. Write the GCF and remaining polynomial as a
product.
***Alert – If the term of the polynomial is exactly the same as the GCF, when you
divide, you are left with 1, not zero.
Example 3:
𝑥(3𝑥 − 1) + 5(3𝑥 − 1)
What is the GCF? Factor it out of the polynomial.
Factoring a Polynomial with 4 Terms
By Grouping
Sometimes there is not a GCF for all the terms in a polynomial. If you have 4 terms,
try grouping them.
a.
Group the 1st 2 terms together and the last 2 terms together.
b.
Factor out a GCF for the first set and a GCF from the second set.
c.
Factor out a common binomial if possible.
Example 4:
𝑥 3 + 3𝑥 2 + 2𝑥 + 6
Example 5:
7𝑥 3 − 14𝑥 2 − 𝑥 + 2
Factoring Trinomials Where
The Leading Coefficient is 1
𝑥 2 + 𝑏𝑥 + 𝑐
To factor this quadratic expression, students need to find the 2 x-terms that multiply
together to equal 𝑥 2 times 8. (The product of the diagonals of the rectangle should
be equal.) Also, the sum of the 2 x-terms is b. Check this concept by examining the
rectangle at the top of this page.
Example 6:
Find the 2 terms whose product is 12𝑥 2 and whose sum is 7𝑥 . Write these
terms in the other diagonal. It doesn’t matter which one goes where.
Factor a GCF out of each row and column. (Find the dimensions of each side of
the rectangle.) Write the completed factored form.
Example 7:
Factor
𝑥 2 + 7𝑥 − 30
Example 8:
Factor
𝑥 2 − 15𝑥 + 56
Factoring Trinomials of the Form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Use the same method as before – a rectangle model.
Example 9:
Factor the GCF for each row and column. Check the signs of each factor. Write the
completed factored form.
Example 10:
Factor
3𝑥 2 + 21𝑥 + 36
Note: If a common factor appears in all the terms, it should be factored out first.
Then follow the previous examples.
Factoring a Perfect Square Trinomial
𝑎2 + 2𝑎𝑏 + 𝑏 2 = (𝑎 + 𝑏)2
OR
𝑎2 − 2𝑎𝑏 + 𝑏 2 = (𝑎 − 𝑏)2
The trinomial must be in this exact form. If you aren’t sure, you can use the general
process for factoring.
Example 11:
𝑦 2 + 14𝑦 + 49
Example 12:
9𝑥 2 − 30𝑥𝑦 + 25𝑦 2
Factoring a Difference of Two Squares
𝑥 2 − 𝑎2 = (𝑥 + 𝑎)(𝑥 − 𝑎)
***Alert – the sum of 2 perfect squares does not factor.
The difference of 2 squares must be in exactly this form.
Example 13:
𝑦 2 − 16
Example 14:
25𝑥 2 − 81
Factoring a Sum of Two Cubes
Factoring a Difference of Two Cubes
𝑥 3 + 𝑎3 = (𝑥 + 𝑎)(𝑥 2 − 𝑎𝑥 + 𝑎2 )
𝑥 3 − 𝑎3 = (𝑥 − 𝑎)(𝑥 2 + 𝑎𝑥 + 𝑎2 )
To use this rule, the polynomial must be in one of these exact forms.
Example 15:
64𝑎3 + 1
Example 16:
27𝑥 3 − 8
Put it all Together:
Examples of Many Kinds
Example 17:
27𝑎2 + 36𝑎 + 12
Example 18:
𝑦 4 − 16
Example 19:
𝑥 3 + 64𝑦 3
Example 20:
6𝑥 2 − 17𝑥 + 5
Example 21:
𝑥 2 + 5𝑏𝑥 − 2𝑎𝑥 − 10𝑎𝑏