Keeper 1 Honors Calculus Factoring Keeper 1 Honors Calculus

What does it mean to "factor?" Take an addition or subtraction problem and write it as an EQUIVALENT multiplication problem.

To factoring using GCF: Step 1: Determine the greatest common factor of the given terms. The greatest common factor or GCF is the largest factor that all terms have in common. Step 2: Factor out (or divide out) the greatest common factor from each term. ***You could check your answer at the point by distributing the GCF to see if you get the original question. ***Factoring out the GCF is the first step in many factoring problems.

Example: Factor using GCF. 12 𝑥 3 +16

Example: Factor using GCF. 9 𝑥 3 −27𝑥+36

Example: Factor using GCF. 2𝑥+5

Factoring Quadratic Trinomials where 𝑎=1 Identify the a, b, and c in the trinomial 𝑎 𝑥 2 +𝑏𝑥+𝑐. Identify two numbers that multiply to 𝑐 and add to 𝑏 Substitute the two numbers into two binomials. Write your answer as: (x ___)(x ___)   There are only 4 possible sign combinations for these trinomials. Don't forget to ALWAYS start by looking for a GCF!

Example: Factor 𝑥 2 +10𝑥+24

Example: Factor 𝑥 2 −5𝑥−14

Example: Factor 𝑥 2 −7𝑥+10

Example: Factor 𝑥 2 +𝑥−20

Putting it all together! Example: Factor 5 𝑥 2 +35𝑥+60

How to Factor when 𝑎≠1 1.) Multiply "𝑎" and "𝑐" 2.) Determine what 2 numbers multiply to "𝑎𝑐" that sum to "𝑏" 3.) Rewrite "𝑏" as those 2 numbers (include the variable) You should now have 4 terms! 4.) Factor by grouping. *Put (first 2 terms)(last 2 terms) *Factor out a GCF from each parenthesis. GCF(leftovers) +/- GCF(leftovers) *Your leftovers should be identical! *Write your factors as (GCF's)(leftover) Remember to always start by checking for a GCF of the ENTIRE problem first!

Example: Factor 2 𝑥 2 +11𝑥+12

Example: Factor 8 𝑥 2 +2𝑥−3

Example: Factor −5 𝑥 2 −7𝑥+6

Difference of Squares We use DOS when you have… 𝑎 2 − 𝑏 2 *2 Terms *1 Minus Sign *Coefficients and Constants are Perfect Squares *Variables have even exponents

Difference of Squares Take the square root of each term. Write your answer as: (𝑎+𝑏)(𝑎−𝑏)

Example: Factor 𝑥 2 −25

Example: Factor 4 𝑥 2 −49

You Try!!! 30 𝑎 2 −111𝑎+90

You Try!!! 10 𝑝 2 +57𝑝+54

You Try!!! 18 𝑚 2 −50𝑚𝑛−12 𝑛 2

You Try!!! 9 𝑥 2 +4

You Try!!! 169 𝑣 4 −1

You Try!!! 1452 𝑛 8 −1728 𝑛 2

You Try!!! −8 𝑥 3 +64

Sums and Difference of Cubes

Example: Factor 𝑥 3 +125

Example: Factor 8 𝑥 3 −27

Example: Factor 2𝑥 3 +128 𝑦 3