Download Factoring Patterns

Factoring Guide
Algebra II
1.
Name_________________
GCF – Always look for the Greatest Common Factor
2x3 + 4x2
2x2(x + 2)
- Find the greatest common factor for all terms
Factoring Patterns
A. Difference of two squares
Example
(a2 - b2) = (a – b)(a + b)
(16x2 – 81)
4x
9
- Find square root of each term
(4x – 9)(4x + 9)
- Put into the formula (a - b)(a + b)
---------------------------------------------------------------------------------------------------------------------------B. Perfect Square Trinomial
Example
(4x2 + 44x + 121)
2x
11
2 2x 11 = 44x
+
(2x + 11)2
(a2 + 2ab + b2) = (a + b)2 or
(a2 – 2ab + b2) = (a – b)2
- Find square root of the first and last term
- Check to see if 2 times the first and last term
equals the middle term
- Examine sign of middle term
- Put into the formula (a + b)2
---------------------------------------------------------------------------------------------------------------------------C. Sum of Two Cubes
Example
(a3 + b3) = (a2 – ab + b2)
(8x3 + 64)
2x
4
(2x + 4)(4x2 – 8x + 16)
- Find the cube root of each term
- Put into the equation (a + b)(a2 – ab + b2)
---------------------------------------------------------------------------------------------------------------------------D. Difference of Two Cubes
Example
(a3 - b3) = (a2 + ab + b2)
(27x3 - 125)
3x
5
(3x - 5)(9x2 + 15x + 25)
- Find the cube root of each term
- Put into the equation (a - b)(a2 + ab + b2)
----------------------------------------------------------------------------------------3. Grouping there are 4 terms
Example
4x3 - 12x2 - 6x + 18
(4x3 - 12x2) + (- 6x + 18)
4x2(x – 3) + -6(x - 3)
- Group the first two terms together and the
last two terms together
- Find the GCF for each group and factor out
(4x2 – 6)(x - 3)
- Use distributive property
-------------------------------------------------------------------------------------------------------4. Simple Factoring in the form of x2 + bx + c
Example
x2 + 4x - 5
-1 5 or
1 -5
(x + )(x - )
- Factor the last term
- Look at last term. If negative then signs in the
factored terms are different. If positive then signs in
factored terms are the same.
(x + 5)(x – 1)
- Find the two factors of the last term that
add up to the middle term
-------------------------------------------------------------------------------------------------------5. Difficult Trinomial in the form of ax2 + bx + c
Example
4x2 + 5x - 9
4
-9 = -36
-1 36
-2 18
-3 12
-4 9
-6 6
1
2
3
4
6
9
- Multiply first and last coefficients
-36
-18
-12
-9
-6
-4
- Factor the product
- Find the product that adds up to middle term
(Since last coefficient is negative, one term
must be positive the other negative)
4x2 + 9x – 4x – 9
- Rewrite the original quadratic with the
middle term broken up into two middle terms
x(4x + 9) – (4x + 9)
- Factor by grouping
(x – 1) (4x + 9)
-------------------------------------------------------------------------------------------------------Lets put it all together now.
Example:
2x5 + 24x = 14x3
2x5 − 14x3 + 24x = 0
Set = to zero in preparation for the
Zero Product Property
2x(x4 − 7x2 + 12 ) = 0
Common Monomial.
2x(x2 − 3)(x2 − 4) = 0
Easy trinomial
2x(x2 − 3)(x + 2)(x − 2) = 0
A little more factoring!
2x = 0
x = 0
x2 − 3 = 0
x =  3
x + 2 = 0
x = −2
x − 2 = 0
Set all factors to 0.
x = 2
The solutions are 0, 3 ,  3 , −2, 2. Wow!! (By the way, notice that there are five solutions and the
degree is 5 . . . hmmmm)
Algebra II
5.4 Factoring and Solving Polynomial Equations Name___________________Date _____
Classwork
Solve.
1. 125x3 + 8 = 0
2. x3 – 3x2 = 0
3.
3x4 + 15x2 – 72 = 0
5. 2x4 + 128x = 0
4.
2x5 - 12x3 = -16x
6. x3 - 3x2 – 16x + 48 = 0