1. 5.1 Factoring – the Greatest Common Factor
Finding the Greatest Common Factor:
Factor – write each number in factored form.
List common factors
Choose the smallest exponents – for variables and prime factors
Multiply the primes and variables from step 3
Always factor out the GCF first when factoring an expression

2. 5.1 Factoring – the Greatest Common Factor
Example: factor 5x2y + 25xy2z

3. 5.1 Factoring – Factor By Grouping
Factoring by grouping
Group Terms – collect the terms in 2 groups that have a common factor
Factor within groups
Factor the entire polynomial – factor out a common binomial factor from step 2
If necessary rearrange terms – if step 3 didn’t work, repeat steps 2 & 3 until you get 2 binomial factors

4. 5.1 Factoring – Factor By Grouping
Example: This arrangement doesn’t work.
Rearrange and try again

5. 5.2 Factoring Trinomials
Factoring x2 + bx + c (no “ax2” term yet) Find 2 integers: product is c and sum is b
Both integers are positive if b and c are positive
Both integers are negative if c is positive and b is negative
One integer is positive and one is negative if c is negative

6. 5.2 Factoring Trinomials
Example:

7. 5.3 Factoring Trinomials – Factor By Grouping
Factoring ax2 + bx + c by grouping
Multiply a times c
Find a factorization of the number from step 1 that also adds up to b
Split bx into these two factors multiplied by x
Factor by grouping (always works)

8. 5.3 Factoring Trinomials – Factor By Grouping
Example:
Split up and factor by grouping

9. 5.3 More on Factoring Trinomials
Factoring ax2 + bx + c by using FOIL (in reverse)
The first terms must give a product of ax2 (pick two)
The last terms must have a product of c (pick two)
Check to see if the sum of the outer and inner products equals bx
Repeat steps 1-3 until step 3 gives a sum = bx

10. 5.3 More on Factoring Trinomials
Example:

11. 5.3 More on Factoring Trinomials
Box Method (not in book):

12. 5.3 More on Factoring Trinomials
Box Method – keep guessing until cross-product terms add up to the middle value

13. 5.4 Special Factoring Rules
Difference of 2 squares:
Example:
Note: the sum of 2 squares (x2 + y2) cannot be factored.

14. 5.4 Special Factoring Rules
Perfect square trinomials:
Examples:

15. 5.4 Special Factoring Rules
Difference of 2 cubes:
Example:

16. 5.4 Special Factoring Rules
Sum of 2 cubes:
Example:

17. 5.4 Special Factoring Rules
Summary of Factoring
Factor out the greatest common factor
Count the terms:
4 terms: try to factor by grouping
3 terms: check for perfect square trinomial. If not a perfect square, use general factoring methods
2 terms: check for difference of 2 squares, difference of 2 cubes, or sum of 2 cubes
Can any factors be factored further?

18. 5.5 Solving Quadratic Equations by Factoring
Zero-Factor Property: If a and b are real numbers and if ab=0 then either a = 0 or b = 0

19. 5.5 Solving Quadratic Equations by Factoring
Solving a Quadratic Equation by factoring
Write in standard form – all terms on one side of equal sign and zero on the other
Factor (completely)
Set all factors equal to zero and solve the resulting equations
(if time available) check your answers in the original equation

20. 5.5 Solving Quadratic Equations by Factoring
Example:

21. 5.6 Applications of Quadratic Equations
This section covers applications in which quadratic formulas arise. Example: Pythagorean theorem for right triangles (see next slide)

22. 5.6 Applications of Quadratic Equations
Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2 c a b

23. 5.6 Applications of Quadratic Equations
Example
x+2 x
x+1