Multiplying Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve the problem step by step, starting with filling in the definitions and then solving the multiplication problems.

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Part 1: Fill in the Blanks

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

✔ Answer: *terms*

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

✔ Answer: *one*

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

✔ Answer: *two*

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

✔ Answer: *each*

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

Distribute $2x$ to each term:

- $2x \cdot 6x^4 = 12x^{1+4} = 12x^5$
- $2x \cdot x^3 = 2x^{1+3} = 2x^4$

✔ Answer: $12x^5 + 2x^4$

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#### 2. $y(7y^5 + 8y^2)$

Distribute $y$:

- $y \cdot 7y^5 = 7y^{1+5} = 7y^6$
- $y \cdot 8y^2 = 8y^{1+2} = 8y^3$

✔ Answer: $7y^6 + 8y^3$

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

Distribute $8x^3$:

- $8x^3 \cdot 4x^5 = 32x^{3+5} = 32x^8$
- $8x^3 \cdot (-xy) = -8x^{3+1}y = -8x^4y$
- $8x^3 \cdot 2x = 16x^{3+1} = 16x^4$

✔ Answer: $32x^8 - 8x^4y + 16x^4$

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

Distribute $-9x^3$:

- $-9x^3 \cdot (-2x^5) = 18x^{3+5} = 18x^8$
- $-9x^3 \cdot y = -9x^3y$
- $-9x^3 \cdot (-4x) = 36x^{3+1} = 36x^4$

✔ Answer: $18x^8 - 9x^3y + 36x^4$

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YOU TRY:

Now try these on your own — here are the solutions.

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

Distribute $-3x^4$:

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

✔ Answer: $-24x^8 + 6x^{11}$

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

Distribute $5y^2$:

- $5y^2 \cdot 4x^3y = 20x^3y^{2+1} = 20x^3y^3$
- $5y^2 \cdot x^2y = 5x^2y^{2+1} = 5x^2y^3$
- $5y^2 \cdot (-2xy^3) = -10xy^{2+3} = -10xy^5$

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

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✔ Final Answers Summary:

#### Definitions:
- Polynomial: terms
- Monomial: one
- Binomial: two
- Distributive property: each

#### 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. $20x^3y^3 + 5x^2y^3 - 10xy^5$

Let me know if you'd like this formatted as a completed worksheet!