1. Flowchart to factor Factor out the Great Common Factor
Ex 1
Can be the expression
written as A2 – B2 or
A+ B3 or A3 – B3 ?
YES
Ex 2
Factor
Two terms?
YES NO
END NO
Can the expression be
written as ax2+bx+c?
YES
YES
perfect square?
Ex 3
Three Terms?
YES NO NO
product of two
binomials?
YES
END NO
Ex 4 NO
END
Can you factorize the
expression by grouping ?
Four or More
Terms?
YES
YES
Ex 5 NO
END

2. Ex 1: Greatest Common Factor (GCF)
Ex 1a: Factor 6ab+8ac+4a
Look for the GCF ab+8ac+4a = 2 ·3 ab + 2 · 4ac + 2 ·2a
Take out the GCF = 2a (      )
Working on the parenthesis = 2a ( 3 b c )
Answer: 6ab+8ac+4a = 2a ( 3b + 4c + 2)
Ex 1b: Factor 5x2y(2a–3) –15xy(2a-3)
Look for the GCF 5x2y(2a–3)–15xy(2a-3) = 5xxy(2a-3)– 3•5xy(2a-3)
Take out the GCF = 5xy(2a-3)(  )
Working on the parenthesis = 5xy(2a-3)(x )
Answer: 5x2y(2a – 3) –15xy(2a - 3) = 5xy(2a - 3)( x - 3 )
Return to Flowchart

3. Ex 2: Factoring a Binomial
Ex 2a: Factor 3x3 – 12xy2
Taking the GCF out x3 – 12xy2 = 3x ( x2 – 4y2 )
Factoring as difference of squares
= 3x (x+2y)(x -2y)
Answer: 3x3 – 12xy2 = 3x (x+2y(x -2y)
Ex 2b: Factor 16x4 + 54x2
Taking the GCF out x5 + 54x2 = 2x2 (8x )
Working on the parenthesis = 2x2( [2x]3 + [3]3 )
Factoring as difference of cubes
= 2x2 (2x+ 3)[ (2x)2 –(2x)3 +(3)2 ]
Simplifying = 2x2 (2x+ 3 )(4x2 - 6x + 9)
Answer : 16x5 – 54x2 = 2x2(2x + 3)(4x2 – 6x + 9)
Return to Flowchart

4. Ex 3: Factoring as Perfect Square
Ex 3a: Factor 63x4 – 210x x2
Taking GCF out x4 – 210x x2 = 7x2 (9x2 – 30x + 25)
Check for perfect square (    )2
Take square root first and last terms = 7x2 (3x – )2
Middle term’s check (3x)5 = 30x OK!
Answer: 63x4 – 30x3 + 25x2 = 7x2 (3x – 5 )2
Ex 3b: Factor 3x3y2 +18x2y +27x
Taking the GCF out x3y2 + 18x2y +27x = 3x (x2y2 + 6xy + 9)
Check for perfect square (    )2
Take square root first and last terms = 3x ( xy )2
Middle term’s check (xy)3 = 6xy OK!
Answer: 3x3y2 +18x2y + 27 =3x (xy + 5)2
Return to Flowchart

5. Ex 4: Factor as Product of Two Binomials Simple Case
Ex 4a : Factor 3x3 – 21x2 + 30x
Taking GCF out x3 – 21x2 + 30x = 3x (x2 – 7x+ 10)
Trinomial is not a perfect square
Let’s try to factor as product of two binomials = 3x(x + a )(x + b )
Where a·b = 10 & a + b = - 7. Since a·b is positive, a & b should have same sign, and since ab = both should be negative.
Possibilities for a, b are 1,10 or 2,5. Let’s check x x
- x
- 10x
-11x
x - 2
x - 5
-2x
-5x
-7x
NO!
OK!
So, a = -2 and b = x3 – 21x2 + 30x = 3x (x -2)(x –5)
Answer: 3x3 – 21x2 + 30x = 3x (x - 2)(x - 5)
Return to Flowchart

6. Ex 4: Simple Case Continued …
Ex 4b: Factor 4x5 – 20x3 + 16x
Taking the GCF out x5 – 20x3 + 16x = 4x (x4 – 5x2 + 4 )
Trinomial is not a perfect square !  
Factoring as product of two binomials = 4x (x2 – 1) (x2 – 4)
Factoring as difference of squares    
=4x (x+1)(x–1)(x+2)(x–2)
Answer: 4x5– 20x3+16x = 4x (x+1)(x–1)(x+2)(x–2)
Ex 4c: Factor 128 x7 – 2x
Taking the GCF out … x7 – 2x = 2x (64x6 – 1)
Working on the parenthesis = 2x ( (8x3 )2 – 1 )
 
Factoring as difference of squares = 2x (8x3 + 1) (8x3 – 1)
Working on the parenthesis … = 2x ( (2x)3+ 1 ) ( (2x)3 –1 )
   
Addition and difference of cubes =2x(2x+1)(4x2 –2x+1)(2x–1)(4x2+2x+1)
Answer: 128 x7–162x =2x(2x+1)(4x2–2x+1)(2x-1)(4x2+2x+1)

7. Ex 4: General Case 7 Ex 4d: Factor 30 x2 – 21x – 36
Taking the GCF out x2 – 21x – 36 = 3(10x2 – 7x – 12)
Check if it factors as = 3(ax + p)(bx + q)
Since ab = 10, possible values for a,b are 1&10, or 2&5
Since pq = - 12, p & q has different sign and all possible ways of getting 12 are 12(1) , 6(2), 4(3).
The following tables summarize the situation for pairs 1&10, 2&5
119
- 2 58 14 26 37 1 12 6 2 3 4 10
- 1
-12
- 2
- 6
- 4
- 3 2 12 1 6 3 4 5
- 1
-12
- 2
- 6
- 4
- 3 58
-19 26 14 2 12 1 6 3 4 5
- 1
-12
- 2
- 6
- 4
- 3 7
Each number on the last row is the adding of cross multiplication of the blue column by each one of the red columns. For example: 1 (-1) + 10 (12) = 119 …
The pair 1 & 10 doesn’t work ! ( since 7 didn’t appear on the last row).
Let’s try now with p=2 & q=5. Filling the blanks on third row …
Since we need – 7x as the middle term, so we pick p = -2 and q = 5. (Do the Check)
Answer: 30x2 – 21x – 36 = 3(2x - 3)(5x +4)
Return to Flowchart

8. Ex 5: Factor by grouping Ex 5a: Factor 3x3 – 6x2 + 5x –10
Splitting in two groups x3 – 6x2 + 5x –10 = 3x3 – 6x x –10
Taking the GCF out from both groups = 3x2 (x – 2) + 5 (x – 2)
Taking the GCF ( x – 2 ) out (to the right)
= ( 3x2 + 5)
Answer: 3x3 – 6x2 + 5x –10 = ( x – 2)( 3x2 + 5)
Ex 5b: Factor 4x2 – 40x – 4y2
Taking the GCF out x2 – 40x – 4y2 = 4(x2 – 10x + 25 –y2)
Split expression in parenthesis into two groups = 4 ([x2–10x+25]– y2 )
Factor the first group as perfect square = 4 ([ x ]2 – y2 )
Factor as difference of squares
( x – 2)
= 4 ( [x – 5 + y][x – 5 – y] )
Answer: 4x2 – 40x –4y2 = 4(x + y – 5)(x – y – 5)
Return to Flowchart