1. Section 9.6 What we are Learning:
To multiply a polynomial by a monomial
To simplify expressions involving polynomials

2. Distributive Property:
For any real number
a(b + c) = ab + ac
This means: Multiply everything inside of the grouping symbol by what is on the outside of the grouping symbol.

3. Multiplying a Polynomial by a Monomial:
Combine any like terms first
Use the distributive property: Multiply everything inside of the grouping symbol by what is on the outside of the grouping symbol.
Use your Properties of Powers to multiply variables with exponents
You can multiply vertically or horizontally

4. Example: -5x3(7x2 + 8x -3) Vertically Horizontally -5x3(7x2 + 8x -3)
-5x3 (7x2) + (-5x3)(8x) + (-5x3)(-3)
-35x5 - 40x4 + 15x3
-5x3(7x2 + 8x -3)
7x2 + 8x -3
(x) x3
-35x5 - 40x4 + 15x3

5. Example: Find 3x2(6x2y + 4 – 2x2y)
Vertically
Horizontally
3x2(6x2y + 4 – 2x2y)
3x2(4x2y + 4)
3x2(4x2y) + 3x2(4)
12x4y + 12x2
3x2(6x2y + 4 – 2x2y)
3x2(4x2y + 4)
4x2y + 4
(x) x2
12x4y + 12x2

6. Solving Equations That Contain Polynomials:
Combine any like terms
Use the Distributive Property
Use the Addition/Subtraction, Multiplication/Division Properties of Equality to solve for the variable.
Remember: An equation is solved when a variable with a coefficient of one is on one side of the equation and the solution is on the other.

7. Example: 2x(x – 4) + 8x – 3 = 2x(5 + x) – 5x + 12
There are no like terms
[2x(x) + 2x(-4)] + 8x – 3 = [2x(5) + 2x(x)] – 5x + 12
Distributive Property
2x2 – 8x + 8x – 3 = 10x + 2x2 – 5x + 12
Combine like terms
2x2 – 3 = 5x + 2x2 + 12
2x2 – 2x2 – 3 = 5x + 2x2 – 2x2 + 12
Subtraction Property of equality
-3 = 5x + 12
-3 – 12 = 5x + 12 – 12
-15 = 5x
-15/5 = 5x/5
Division Property of equality
-3 = x
Solution

8. Let’s Work These Together: Find each product
4a2(-8a3c + c -11)
11ab(2ab – 5a)

9. Let’s Work These Together:
Simplify
Solve
8m(-9m2 + 2m – 6) +
11(2m2 – 4m + 12)
y(y + 12) – 8y = 14 + y(y – 4)

10. Homework:
Page 532
17 to 21 odd
27 to 31 odd
33 to 37 odd