Summary of Factoring Techniques

1. Factor out the Greatest Common Factor (G.C.F.) Always look for this first!

Example 1: 6x³ + 12x² = 6x²(x + 2)
Example 2: 28x⁴y² + 14x²y = 14x²y( 2x²y + 1)

2. Factor by Grouping This applies if the polynomial has 4 terms .

Ex: 2y² − 6y + 3xy − 9x = 2y(y − 3) + 3x(y − 3) = (y − 3)(2y + 3x)

3. Use a Special Formula, if applicable. There are special factoring formulas for some special cases. They are the difference of 2 perfect squares, the sum of 2 perfect cubes and the difference of 2 perfect cubes. Note: the sum of 2 perfect squares does not factor!

Formulas: a² + b² = not factorable (prime)
a² − b² = (a + b)(a − b)
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

Examples: 49 − z² = (7 − z)(7 + z)
3y³ + 648 = 3(y³ + 216) = 3((y)³ + (6)³) = 3(y + 6)(y² − 6y + 36)
x³ − 64 = (x)³ − (4)³ = (x − 4)(x² + 4x + 16) Hint: see below and use a = x, b= 4
Pattern for a³ + b³ is : (a sign b ) (a² sign ab sign b² )
Signs: (a sign b) is same as original expression , ab is opposite of that, b² is always +

4. General Trinomials Three term polynomials . The product of 2 binomials is generally a trinomial except for the situations in #3 above.

Example: (x + 3) ( 2x + 7) = 2x² + 13x + 21

Factoring trinomials is just the reverse of the FOIL process.
In many instances, using “guess and check” method you can determine the binomial factors. Use the FOIL method to check your guess.
If the leading term's coefficient is 1, you can usually make a good guess.

Example: x² + 3x − 10 = (x − 2)(x + 5) , factor pairs for 10 are 5 x ², and 10 x 1

(Note that in this case we have − 10, which factors into (− 2)(+ 5 ), or (− 5)(+ 2) , but
since (−2) + (+ 5) = +3, we choose those as our factors for the binomial. ) After deciding what to use for the factors, FOIL it out to make sure it gives the original trinomial!

Some students do not like the approach of guessing and checking, in which case a “method” is available.
See the “box” method on the back of this sheet for instruction in that method.

Factoring quadratic trinomials by the “Box” method

Make sure the polynomial is in written in proper order and that the leading quadratic term is positive. Sometimes this requires factoring out a − 1 or other common factor. Example: −2x² − 10x + 12 is −2( x² + 5x − 6)

Out line :
1)Start with a standard “Tic-Tac-Toe” board, crossing out the upper left corner
2) Fill in middle and lower right boxes with first & last terms of trinomial.
3) Find the Key Product (of filled-in boxes) and factor it all
possible ways until you find a pair of factors that would add up to the middle term of the trinomial.
4) Fill in the other two boxes with those factors.
5) Factor out the GCF from each row (sign agrees with middle column)
6) Factor out the common factor from each column (sign agrees with middle row)
7) The factors of your trinomial are in the top row and left column
8) Write out the answer in (a + b)(c + d) format.
9) Check by F.O.I.L.

Example: Factor 6x² − 19x + 10 = _________________ (note BL = BLANK)

Diagonal: (6x²)(10) = 60x² No reason to try any of the positive combinations since middle term, − 19x, is negative

The binomial factors are: (3x – 2)(2x – 5) i.e. the binomial factors in the left column and top row.

Note: when looking at the row with − 15x and + 10 always factor out the negative common factor , not the positive. So, here, we factored out − 5 instead of the +5. We did this again in the far right corner by factoring out the −2. If middle of bottom row or middle of right column is negative, that is when you do this.