1. 7.1 The Greatest Common Factor and Factoring by Grouping
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. 7.1 The Greatest Common Factor and Factoring by Grouping
Example: factor 5x2y + 25xy2z

3. 7.1 The Greatest Common Factor and 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. 7.1 The Greatest Common Factor and Factoring by Grouping
Example: This arrangement doesn’t work.
Rearrange and try again

5. 7.2 Factoring Trinomials of the Form x2 + bx + c
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. 7.2 Factoring Trinomials of the Form x2 + bx + c
Example:

7. 7.3 Factoring Trinomials of the Form ax2 + bx + c
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. 7.3 Factoring Trinomials of the Form ax2 + bx + c
Example:
Split up and factor by grouping

9. 7.3 Factoring Trinomials of the Form ax2 + bx + c
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. 7.3 Factoring Trinomials of the Form ax2 + bx + c
Example:

11. 7.3 Factoring Trinomials of the Form ax2 + bx + c
Box Method (not in book):

12. 7.3 Factoring Trinomials of the Form ax2 + bx + c
Box Method – keep guessing until cross-product terms add up to the middle value

13. 7.4 Factoring Binomials and Perfect Square Trinomials
Difference of 2 squares:
Example:
Note: the sum of 2 squares (x2 + y2) cannot be factored.

14. 7.4 Factoring Binomials and Perfect Square Trinomials
Examples:

15. 7.4 Factoring Binomials and Perfect Square Trinomials
Difference of 2 cubes:
Example:

16. 7.4 Factoring Binomials and Perfect Square Trinomials
Sum of 2 cubes:
Example:

17. 7.4 Factoring Binomials and Perfect Square Trinomials
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. 7.5 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. 7.5 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. 7.5 Solving Quadratic Equations by Factoring
Example:

21. 7.6 Applications of Quadratic Equations
This section covers applications in which quadratic formulas arise. Example: Pythagorean theorem for right triangles (see next slide)

22. 7.6 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. 7.6 Applications of Quadratic Equations
Example
x+2 x
x+1

24. 9.3 Linear Inequalities in Two Variables
A linear inequality in two variables can be written as: where A, B, and C are real numbers and A and B are not zero

25. 9.3 Linear Inequalities in Two Variables
Graphing a linear inequality:
Draw the graph of the boundary line.
Choose a test point that is not on the line.
If the test point satisfies the inequality, shade the side it is on, otherwise shade the opposite side.

26. 9.4 Systems of Linear Equations in Three Variables
Linear system of equation in 3 variables:
Example:

27. 9.4 Systems of Linear Equations in Three Variables
Graphs of linear systems in 3 variables:
Single point (3 planes intersect at a point)
Line (3 planes intersect at a line)
No solution (all 3 equations are parallel planes)
Plane (all 3 equations are the same plane)

28. 9.4 Systems of Linear Equations in Three Variables
Solving linear systems in 3 variables:
Eliminate a variable using any 2 equations
Eliminate the same variable using 2 other equations
Eliminate a different variable from the equations obtained from (1) and (2)

29. 9.4 Systems of Linear Equations in Three Variables
Solving linear systems in 3 variables:
Use the solution from (3) to substitute into 2 of the equations. Eliminate one variable to find a second value.
Use the values of the 2 variables to find the value of the third variable.
Check the solution in all original equations.