Let’s solve each problem one by one. We’re multiplying a monomial (a single term) by a polynomial (multiple terms). To do this, we use the
distributive property: multiply the monomial by *each* term inside the parentheses.
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Problem 1:
3q(q² - 6q + 5)
→ Multiply 3q by q² → 3q³
→ Multiply 3q by -6q → -18q²
→ Multiply 3q by 5 → 15q
✔ Final:
3q³ - 18q² + 15q
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Problem 2:
9(x² + xy - 8y²)
→ 9 × x² = 9x²
→ 9 × xy = 9xy
→ 9 × (-8y²) = -72y²
✔ Final:
9x² + 9xy - 72y²
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Problem 3:
7(6x² + 9xy + 10y²)
→ 7 × 6x² = 42x²
→ 7 × 9xy = 63xy
→ 7 × 10y² = 70y²
✔ Final:
42x² + 63xy + 70y²
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Problem 4:
3x(2x² - 5x + 8)
→ 3x × 2x² = 6x³
→ 3x × (-5x) = -15x²
→ 3x × 8 = 24x
✔ Final:
6x³ - 15x² + 24x
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Problem 5:
x²(-x² + 2x + 7)
→ x² × (-x²) = -x⁴
→ x² × 2x = 2x³
→ x² × 7 = 7x²
✔ Final:
-x⁴ + 2x³ + 7x²
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Problem 6:
10(-x² + 10x - 6)
→ 10 × (-x²) = -10x²
→ 10 × 10x = 100x
→ 10 × (-6) = -60
✔ Final:
-10x² + 100x - 60
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Problem 7:
2k²(2k³ + 6k - 4)
→ 2k² × 2k³ = 4k⁵
→ 2k² × 6k = 12k³
→ 2k² × (-4) = -8k²
✔ Final:
4k⁵ + 12k³ - 8k²
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Problem 8:
6x²(3x² - 5xy - 6)
→ 6x² × 3x² = 18x⁴
→ 6x² × (-5xy) = -30x³y
→ 6x² × (-6) = -36x²
✔ Final:
18x⁴ - 30x³y - 36x²
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Problem 9:
-x(-x² - 4x + 4xy)
→ -x × (-x²) = x³
→ -x × (-4x) = 4x²
→ -x × 4xy = -4x²y
✔ Final:
x³ + 4x² - 4x²y
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Problem 10:
3r(2r² - 3ry + 2r)
→ 3r × 2r² = 6r³
→ 3r × (-3ry) = -9r²y
→ 3r × 2r = 6r²
✔ Final:
6r³ - 9r²y + 6r²
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Final Answer:
1. 3q³ - 18q² + 15q
2. 9x² + 9xy - 72y²
3. 42x² + 63xy + 70y²
4. 6x³ - 15x² + 24x
5. -x⁴ + 2x³ + 7x²
6. -10x² + 100x - 60
7. 4k⁵ + 12k³ - 8k²
8. 18x⁴ - 30x³y - 36x²
9. x³ + 4x² - 4x²y
10. 6r³ - 9r²y + 6r²
Parent Tip: Review the logic above to help your child master the concept of monomial and polynomial worksheet.
