1. Distribute each problem:
NEW SEATS!
M3U4D1 Warm-UP
Distribute each problem:

2. M3U4D1 Evaluating and Operating with Polynomials
OBJ: To review adding, subtracting, multiplying, and factoring polynomials

3. How do I evaluate polynomial functions?
You have 3 minutes to complete the top of handout page 1.
Discuss

4. How do I operate with polynomial functions?
Let’s review…

5. The sum f + g Combine like terms & put in descending order
This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.
Combine like terms & put in descending order

6. The difference f - g Distribute negative
To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms.
Distribute negative

7. The product f • g
To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function.
FOIL
Good idea to put in descending order

8. The quotient f /g
To find the quotient of two functions, put the first one over the second.
Nothing more you could do here. (If you can reduce these you should. More later…)

9. Operations with Polynomials
Now you try the four operations on the bottom of page 1 of your handout.

10. Factoring Review:
#1: GCF Method

11. GCF Method is just
distributing backwards!!

12. Review: What is the GCF of 25a2 and 15a?
Let’s go one step further…
1) FACTOR 25a2 + 15a.
Find the GCF and divide each term
25a2 + 15a = 5a( ___ + ___ )
Check your answer by distributing. 5a 3

13. 2) Factor 18x2 - 12x3. Find the GCF 6x2 Divide each term by the GCF
18x2 - 12x3 = 6x2( ___ - ___ )
Check your answer by distributing. 3 2x

14. 3) Factor 28a2b + 56abc2. GCF = 28ab Divide each term by the GCF
28a2b + 56abc2 = 28ab ( ___+ ___)
Check your answer by distributing.
28ab(a + 2c2) a
2c2

15. 4) Factor 20x2 - 24xy x(20 – 24y) 2x(10x – 12y) 4(5x2 – 6xy)

16. 5) Factor 28a2 + 21b - 35b2c2 4a2 3b 5b2c2 GCF = 7
Divide each term by the GCF
28a2 + 21b - 35b2c2 = 7 ( ___ + ___ - ____ )
Check your answer by distributing.
7(4a2 + 3b – 5b2c2)
4a2 3b
5b2c2

17. Factor 16xy2 - 24y2z + 40y2 2y2(8x – 12z + 20) 4y2(4x – 6z + 10)
8xy2z(2 – 3 + 5)

18. Greatest Common Factors aka GCF’s
Factor out the GCF for each polynomial: Factor out means you need the GCF times the remaining parts.
a) 2x + 4y
5a – 5b
18x – 6y
2m + 6mn
5x2y – 10xy
2(x + 2y)
How can you check?
5(a – b)
6(3x – y)
2m(1 + 3n)
5xy(x - 2)

19. Ex 1
15x2 – 5x
GCF = 5x
5x(3x - 1)

20. Ex 2
8x2 – x
GCF = x
x(8x - 1)

21. Ex 3
8x2y4+ 2x3y5 - 12x4y3
GCF = 2x2y3
2x2y3 (4y + xy2 – 6x2)

22. #2: X-box Factoring aka Diamond Method

23. X- Box
Product 3 -9
Sum

24. X-box Factoring
This is a guaranteed method for factoring quadratic equations—no guessing necessary!
We will review how to factor quadratic equations using the x-box method
Background knowledge needed:
Basic x-solve problems
General form of a quadratic equation

25. Factor the x-box way Example: Factor 3x2 -13x -10 (3)(-10)= -30 x -5
-15 2
-13 2x
-10 +2
3x2 -13x -10 = (x-5)(3x+2)

26. First and Last Coefficients
Factor the x-box way
y = ax2 + bx + c
Base 1
Base 2
Product
ac=mn
First and Last Coefficients
1st Term
Factor n
GCF n m
Middle
Last term
Factor m
b=m+n
Sum
Height

27. Examples Factor using the x-box method. 1. x2 + 4x – 12 x +6 x2 6x x
a) b) x +6
-12 4
x2 6x
-2x -12 x 6 -2 -2
Solution: x2 + 4x – 12 = (x + 6)(x - 2)

28. Examples continued 2. x2 - 9x + 20 x -4 x x2 -4x -5x 20
a) b) x -4 20 -9 x
x2 -4x
-5x 20 -5
Solution: x2 - 9x + 20 = (x - 4)(x - 5)

29. Examples continued 3. 2x2 - 5x - 7 2x -7 x 2x2 -7x 2x -7
a) b) 2x -7
-14 -5 x
2x2 -7x
2x -7 +1
Solution: 2x2 - 5x – 7 = (2x - 7)(x + 1)

30. Examples continued 3. 15x2 + 7x - 2 3x +2 5x 15x2 10x -3x -2
a) b) 3x +2
-30 7 5x
15x2 10x
-3x -2 -1
Solution: 15x2 + 7x – 2 = (3x + 2)(5x - 1)

31. #3: Difference of Squares
a2 – b2 = (a + b)(a - b)

32. What is a Perfect Square
Any term you can take the square root evenly (No decimal) 25 36 1 x2 y4

33. Difference of Perfect Squares
x2 – 4 =
the answer will look like this: ( )( )
take the square root of each part:
( x 2)(x 2)
Make 1 a plus and 1 a minus:
(x + 2)(x - 2 )

34. FACTORING Difference of Perfect Squares EX: x2 – 64 How:
Take the square root of each part. One gets a + and one gets a -.
Check answer by FOIL.
Solution:
(x – 8)(x + 8)

35. Example 1
9x2 – 16
(3x + 4)(3x – 4)

36. Example 2
x2 – 16
(x + 4)(x –4)

37. Ex 3
36x2 – 25
(6x + 5)(6x – 5)

38. ALWAYS use GCF first More than ONE Method
It is very possible to use more than one factoring method in a problem
Remember:
ALWAYS use GCF first

39. Example 1
2b2x – 50x
GCF = 2x
2x(b2 – 25)
2nd term is the diff of 2 squares
2x(b + 5)(b - 5)

40. Due back TOMORROW with parent signature
Classwork
M3U4D1 Factoring Review
Evens
Distribute Interims!
Due back TOMORROW with parent signature

41. Due back TOMORROW with parent signature
Distribute Interims!
Due back TOMORROW with parent signature

42. Unit 3 Geometry Test due tomorrow!!!
Homework
M3U4D1 Factoring Review
Odds
Unit 3 Geometry Test due tomorrow!!!
Show all your work to receive credit– don’t forget to check by multiplying!