Multiplying Polynomials Notes worksheet with definitions, examples, and practice problems.

Let's solve the problems step by step and fill in the blanks in the notes section.

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Filling in the Definitions

1. polynomial - an expression with one or more terms that are joined by addition, subtraction, and multiplication
Example: $ 12y^2 + 5y - 3 $

2. monomial - a polynomial with one term
Example: $ 6x^3 $

3. binomial - a polynomial with two terms
Example: $ x^2 + 7 $

4. distributive property - multiplying a term by each term inside the parentheses
Example: $ 3x^2(2x^4 - 5) $

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Multiplying Monomials by Polynomials

We use the distributive property: Multiply the monomial outside the parentheses by each term inside the parentheses.

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#### Problem 1: $ 2x(6x^4 + x^3) $

Distribute $ 2x $ to each term:

- $ 2x \cdot 6x^4 = 12x^{5} $
- $ 2x \cdot x^3 = 2x^4 $

Answer: $ 12x^5 + 2x^4 $

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#### Problem 2: $ y(7y^5 + 8y^2) $

Distribute $ y $:

- $ y \cdot 7y^5 = 7y^6 $
- $ y \cdot 8y^2 = 8y^3 $

Answer: $ 7y^6 + 8y^3 $

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#### Problem 3: $ (8x^3)(4x^5 - xy + 2x) $

Distribute $ 8x^3 $:

- $ 8x^3 \cdot 4x^5 = 32x^8 $
- $ 8x^3 \cdot (-xy) = -8x^4y $
- $ 8x^3 \cdot 2x = 16x^4 $

Answer: $ 32x^8 - 8x^4y + 16x^4 $

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#### Problem 4: $ -9x^3(-2x^5 + y - 4x) $

Distribute $ -9x^3 $:

- $ -9x^3 \cdot (-2x^5) = 18x^8 $
- $ -9x^3 \cdot y = -9x^3y $
- $ -9x^3 \cdot (-4x) = 36x^4 $

Answer: $ 18x^8 - 9x^3y + 36x^4 $

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YOU TRY:

#### 1. $ -3x^4(8x^4 - 2x^7) $

Distribute $ -3x^4 $:

- $ -3x^4 \cdot 8x^4 = -24x^8 $
- $ -3x^4 \cdot (-2x^7) = 6x^{11} $

Answer: $ -24x^8 + 6x^{11} $

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#### 2. $ 5y^2(4x^3y + x^2y - 2xy^3) $

Distribute $ 5y^2 $:

- $ 5y^2 \cdot 4x^3y = 20x^3y^3 $
- $ 5y^2 \cdot x^2y = 5x^2y^3 $
- $ 5y^2 \cdot (-2xy^3) = -10xy^5 $

Answer: $ 20x^3y^3 + 5x^2y^3 - 10xy^5 $

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✔ Final Answers:

#### Multiplying Monomials by Polynomials:
1. $ 12x^5 + 2x^4 $
2. $ 7y^6 + 8y^3 $
3. $ 32x^8 - 8x^4y + 16x^4 $
4. $ 18x^8 - 9x^3y + 36x^4 $

#### You Try:
1. $ -24x^8 + 6x^{11} $
2. $ 20x^3y^3 + 5x^2y^3 - 10xy^5 $

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🔍 Summary of Steps:

- Use the distributive property.
- Multiply the monomial by each term inside the parentheses.
- Apply exponent rules: $ x^a \cdot x^b = x^{a+b} $
- Keep track of signs (positive/negative).
- Combine like terms if needed (not required here).

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