1. 9-2 Multiplying and Factoring
*Using the Distributive Property

2. To simplify a product of monomials (4x)(2x)
Use the Commutative and Associative Properties of Multiplication to group the numerical coefficients and to group like variable;
Find the product of the numbers
Use the properties of exponents to simplify the variable product.
So your answer is (4x)(2x) = 8x2
(4x)(2x) = (4 · 2)(x · x ) =
(4 · 2) = 8
(x · x) = x1 · x1 = x1+1 = x2

3. Multiply powers with the same base:
You can also use the Distributive Property for multiplying powers with the same base when multiplying a polynomial by a monomial. Simplify -4y2(5y4 – 3y2 + 2) -4y2(5y4 – 3y2 + 2) = y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property -20y y2 + 2 – 8y2 = Multiply the coefficients and add the -20y6 + 12y4 – 8y exponents of powers with the same base.
Remember,
Multiply powers with the same base:
35 · 34 = = 39

4. Multiplying powers with the same base.
Simplify each product. a) 4b(5b2 + b + 6) b) -7h(3h2 – 8h – 1) c) 2x(x2 – 6x + 5) d) 4y2(9y3 + 8y2 – 11)
Remember,
Multiplying powers with the same base.
20b3 + 4b2 + 24b
-21h3 + 56h2 + 7h
2x3 -12x2 + 10x
36y5 + 32y4 – 44y2

5. Factoring a Monomial from a Polynomial
Factoring a polynomial reverses the multiplication process. To factor a monomial from a polynomial, first find the greatest common factor (GCF) of its terms.
The GCF is what the
terms all have in
common!
Find the GCF of the terms of:
4x3 + 12x2 – 8x
List the prime factors of each term.
4x3 = 2 · 2 · x · x x
12x2 = 2 · 2 · 3 · x · x
8x = 2 · 2 · 2 · x
The GCF is 2 · 2 · x or 4x.

6. Find the GCF of the terms of each polynomial
Find the GCF of the terms of each polynomial. a) 5v5 + 10v3 b) 3t2 – 18 c) 4b3 – 2b2 – 6b d) 2x4 + 10x2 – 6x
5v3 3 2b 2x

7. Factoring Out a Monomial
To factor a polynomial completely, you must factor until there are no common factors other than 1.
Factor 3x3 – 12x2 + 15x
Step 1
Find the GCF
3x3 = 3 · x · x · x
12x2 = 2 · 2 · 3 · x · x
15x = 3 · 5 · x
The GCF is 3 · x or 3x
Step 2
Factor out the GCF
3x3 – 12x2 + 15x
= 3x(x2) + 3x(-4x) + 3x(5)
= 3x(x2 – 4x + 5)

8. Use the GCF to factor each polynomial
Use the GCF to factor each polynomial. a) 8x2 – 12x b) 5d3 + 10d c) 6m3 – 12m2 – 24m d) 4x3 – 8x2 + 12x
4x(2x-3)
5d(d2 + 2)
6m(m2 -2m -4)
4x(x2 –2x +3)
Try to factor mentally by scanning the coefficients of each term to find the GCF. Next, scan for the least power of the variable.

9. Pg. 463 2-24, 30-38 Evens only! Due Monday!
Homework/Classwork
Pg , 30-38
Evens only! Due Monday!

10. 9-3/9-4 Multiplying Binomials
Using the infamous FOIL method…
also known as DISTRIBUTING!!!

11. Using the Distributive Property
Distribute x + 4
As with the other examples we have seen, we can also use the Distributive Property to find the product of two binomials.
Simplify: (2x + 3)(x + 4)
2x2 + 8x + 3x + 12 =
2x2 + 11x + 12
2x2
+8x
+3x
+12
Now Distribute 2x and 3

12. Multiplying using FOIL
Another way to organize multiplying two binomials is to use FOIL, which stands for,
“First, Outer, Inner, Last”. The term FOIL is a memory device for applying the Distributive Property to the product of two binomials.
Simplify (3x – 5)(2x + 7)
First Outer Inner Last
(3x – 5)(2x + 7)
= 6x x x
= 6x x
The product is 6x2 + 11x - 35

13. Simplify each product.
a) (6h – 7)(2h + 3)
b) (5m + 2)(8m – 1)
c) (9a – 8)(7a + 4)
(y – 3)(y + 3)
(2x-1)2
12h2 +4h - 21
40m2 + 11m - 2
63a2 -20a - 32
y2 + 3y -3y – 9 or y2 – 9
(2x – 1)(2x – 1)
4x2 - 2x - 2x + 1
4x2 - 4x + 1

14. Applying Multiplication of Polynomials.
area of BIG rectangle =
(2x + 5)(3x + 1)
area of little rectangle =
x(x + 2)
area of white region
= area of BIG rectangle – area of black rectangle
(2x + 5)(3x + 1) – x(x + 2) =
6x2 + 15x + 2x + 5 – x2 – 2x =
Combine like terms…
6x2 – x2 + 15x + 2x – 2x =
5x2 + 15x + 5
Find the area of the white region. Simplify.
2x + 5
x + 2
3x + 1 x
Use the Distributive Property to simplify –x(x + 2)
Use the FOIL method to simplify (2x + 5)(3x + 1)

15. Find the area of the shaded region. Simplify.
area of BIG rectangle =
(5x + 8)(6x + 2)
area of little rectangle =
5x(x + 6)
area of white region
= area of BIG rectangle – area of black rectangle
(5x + 8)(6x + 2) - 5x(x + 6)=
30x2 + 10x + 48x + 16 – 5x2 –30x=
Combine like terms…
Answer: 25x2 + 28x + 16
Find the area of the
white region. Simplify.
5x + 8
6x + 2 5x
x + 6

16. Back of THIS worksheet DUE TOMORROW
Classwork/Homework
Back of THIS worksheet
DUE TOMORROW