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**Section 9.6 What we are Learning:**

To multiply a polynomial by a monomial To simplify expressions involving polynomials

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**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.

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**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

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**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

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**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

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**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.

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**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

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**Let’s Work These Together: Find each product**

4a2(-8a3c + c -11) 11ab(2ab – 5a)

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**Let’s Work These Together:**

Simplify Solve 8m(-9m2 + 2m – 6) + 11(2m2 – 4m + 12) y(y + 12) – 8y = 14 + y(y – 4)

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

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