Multiplying Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem step by step. We’re multiplying monomials by polynomials — that means we use the distributive property: multiply the monomial outside the parentheses by *each term* inside the parentheses.

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Problem 1:

2x (6x⁴ + x³)

Multiply 2x by 6x⁴ → 2 × 6 = 12, and x × x⁴ = x⁵ → 12x⁵
Multiply 2x by x³ → 2 × 1 = 2, and x × x³ = x⁴ → 2x⁴

✔ Final: 12x⁵ + 2x⁴

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Problem 2:

y(7y⁵ + 8y²)

Multiply y by 7y⁵ → 1 × 7 = 7, y × y⁵ = y⁶ → 7y⁶
Multiply y by 8y² → 1 × 8 = 8, y × y² = y³ → 8y³

✔ Final: 7y⁶ + 8y³

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Problem 3:

(8x³)(4x⁵ – xy + 2x)

Multiply 8x³ by 4x⁵ → 8×4=32, x³×x⁵=x⁸ → 32x⁸
Multiply 8x³ by –xy → 8×(-1)= -8, x³×x = x⁴, times y → –8x⁴y
Multiply 8x³ by 2x → 8×2=16, x³×x = x⁴ → 16x⁴

✔ Final: 32x⁸ – 8x⁴y + 16x⁴

*(Note: Order doesn’t matter, but usually we write highest degree first or group like terms — here no like terms to combine.)*

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Problem 4:

–9x³(–2x⁵ + y – 4x)

Multiply –9x³ by –2x⁵ → (-9)×(-2)=18, x³×x⁵=x⁸ → 18x⁸
Multiply –9x³ by y → –9x³y → –9x³y
Multiply –9x³ by –4x → (-9)×(-4)=36, x³×x = x⁴ → 36x⁴

✔ Final: 18x⁸ – 9x³y + 36x⁴

*(Again, order can vary — sometimes written as 18x⁸ + 36x⁴ – 9x³y)*

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## YOU TRY Section:

You Try #1:

–3x⁴ (8x⁴ – 2x⁷)

Multiply –3x⁴ by 8x⁴ → (-3)×8 = -24, x⁴×x⁴ = x⁸ → –24x⁸
Multiply –3x⁴ by –2x⁷ → (-3)×(-2)=6, x⁴×x = x¹¹ → +6x¹¹

✔ Final: 6x¹¹ – 24x⁸
*(We often write highest exponent first → so 6x¹¹ – 24x⁸ is fine)*

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You Try #2:

5y²(4x³y + x²y – 2xy³)

Multiply 5y² by 4x³y → 5×4=20, y²×y = y³, keep x³ → 20x³y³
Multiply 5y² by x²y → 5×1=5, y²×y = y³, keep x² → 5x²y³
Multiply 5y² by –2xy³ → 5×(-2)= -10, y²×y³ = y⁵, keep x → –10xy⁵

✔ Final: 20x³y³ + 5x²y³ – 10xy⁵

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Final Answer:

Multiplying Monomials by Polynomials:

1. 2x(6x⁴ + x³) = 12x⁵ + 2x⁴
2. y(7y⁵ + 8y²) = 7y⁶ + 8y³
3. (8x³)(4x⁵ – xy + 2x) = 32x⁸ – 8x⁴y + 16x⁴
4. –9x³(–2x⁵ + y – 4x) = 18x⁸ – 9x³y + 36x⁴

You Try:

1. –3x⁴(8x⁴ – 2x⁷) = 6x¹¹ – 24x⁸
2. 5y²(4x³y + x²y – 2xy³) = 20x³y³ + 5x²y³ – 10xy⁵