# Factoring

** Factoring out a Common Factor:** The first step in factoring any polynomial
is to

look for anything that all the terms have in common and then factor it out using
the

distributive property .

Example: 20y^{2} - 5y^{5} Here, the terms share the common factor 5y^{2} (i.e. 5 is the
largest

number that divides both 20 and 5, and both terms contain the variable y with 2
being

the smallest exponent ). So we factor it out: 20y^{2} - 5y^{5} = 5y^{2}(4 - y^{3})

**Factoring by Grouping:** Factoring by grouping is useful when we encounter
a polynomial

with more than 3 terms.

**Example:** 3x^{3} + x^{2} - 18x - 6

1. First, we group together terms that share a common factor . (3x^{3} + x^{2}) + (-18x
- 6)

The first group shares an x^{2} and the second shares a -6.

2. Factor out the common factor from each grouping. You should have left the
same

expression in each group . x^{2}(3x+1)+(-6)(3x+1) Here that expression is 3x+1

3. Now factor out that expression. (3x + 1)(x^{2} - 6)

** Factoring Trinomials - Reverse FOIL: **There two basic cases that
we’ll encounter:

1. The leading coefficient is a 1. This is the easier of the two cases: x^{2} + bx
+ c All

we need to do here is find two numbers whose product is c and sum is b

**Example:** x^{2} - 7x + 10 = (x + △)(x + △) We need to find two numbers that

multiply to give us +10, but add to give us -7. Well, -5 and -2 do the trick. So

x^{2} - 7x + 10 = (x + (-2))(x + (-5)) = (x - 2)(x - 5)

2. The leading coefficient is not a 1. Things are a little trickier here, but
not much.

Again, it’s just FOIL in reverse.

**Example:**

We need two numbers to fill in for the hearts that will multiply to 3. How about

3 and 1?

3y^{2} + 7y - 20 = (3y +△)(1y + △)

Now we need two numbers to fill in for the triangles that will multiply to -20

AND when we do the INNERS and OUTERS we get 7y. We’ll use the GUESS

and CHECK method to find the two numbers we need.

Let’s try 10 and -2 first:

(3y - 2)(y + 10) = 3y^{2} + 30y - 2y - 20 = 3y^{2} + 27y - 20

That’s not it! Maybe 5 and -4?

(3y + 5)(y - 4) = 3y^{2} - 12y + 5y - 20 = 3y^{2} - 7y - 20

Close, but the sign on the 7 is wrong . Easy to fix - just switch the signs on
the 5

and 4:

(3y - 5)(y + 4) = 3y^{2} + 12y - 5y - 20 = 3y^{2} + 7y - 20 Presto!!

**
Special Factorizations: **Some polynomials are easy to factor because they fit
a

certain mold.

**– Difference of Squares :** F^{2} - L^{2} = (F + L)(F - L)

**Example:** 16x^{2} - 9 = 42x^{2} - 3^{2} = (4x)^{2} - 3^{2} = (4x + 3)(4x - 3)

– Perfect Squares : These are polynomials that factor into (F + L)^{2} or (F - L)^{2}

The pattern we’re looking for here is F^{2} + 2LF + L^{2} or F^{2} - 2LF + L^{2}

**Example:** x^{2} + 6x + 9 = x^{2} + 2·3x + 3^{2} = (x + 3)^{2}

**Example:** y^{2} - 10y + 25 = y^{2} - 2·5y
+ 5^{2}

– Difference of Cubes : F^{3} - L^{3} = (F - L)(F^{2} + LF + L^{2})

**Example:** 2z^{3} - 54 = 2(z^{3} - 27) = 2(z^{3} - 3^{3}) = 2(z - 3)(z^{2} + 3z + 9)

– Sum of Cubes: F^{3} + L^{3} = (F + L)(F^{2} - LF + L^{2})

**Example:** n^{3} + 216 = n^{3} + 6^{3} = (n + 6)(n^{2} - 6n + 36)

Strategy for Factoring:

1. Always factor out the largest common factor first. This will make life easier
for

any further factoring that may need to be done.

2. Look at the number of terms

– Two terms: Is it a difference of squares, difference of cubes or sum of cubes?

– Three terms: Is it a perfect square? Try reverse FOIL.

– Four or more terms: Try factoring by grouping.

3. Always make sure the polynomial is factored COMPLETELY.

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