Question

com software - infinite pre - algebra
multiplying a polynomial and a monomial
find each product.

1. 8x(6x + 6)
2. 7n(6n + 3)
3. 3r(7r - 8)
4. 8(8k - 8)
5. 10a(a - 10b)
6. 2(9x - 2y)
7. 7x(6x + 4y)
8. 4a(8a - 8b)
9. 3n(n² - 6n + 5)
10. 2k³(2k² + 5k - 4)
11. 8r²(4r² - 5r + 7)
12. 3(3v² + 8v - 5)
13. 7(6x² + 9xy + 10y²)
14. 2u(6u² - 9uv + v²)
15. 9(x² + xy - 8y²)
16. 9v²(u² + uv - 5v²)

Explanation:

Step1: Apply distributive property

For each problem, use the distributive property $a(b+c)=ab+ac$ (or $a(b-c)=ab-ac$, $a(b+c-d)=ab+ac-ad$) to multiply the monomial by each term in the polynomial.

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1) $8x(6x + 6)$

Step1: Distribute $8x$ to both terms

$8x \cdot 6x + 8x \cdot 6$

Step2: Simplify each product

$48x^2 + 48x$

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2) $7n(6n + 3)$

Step1: Distribute $7n$ to both terms

$7n \cdot 6n + 7n \cdot 3$

Step2: Simplify each product

$42n^2 + 21n$

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3) $3r(7r - 8)$

Step1: Distribute $3r$ to both terms

$3r \cdot 7r - 3r \cdot 8$

Step2: Simplify each product

$21r^2 - 24r$

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4) $8(8k - 8)$

Step1: Distribute $8$ to both terms

$8 \cdot 8k - 8 \cdot 8$

Step2: Simplify each product

$64k - 64$

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5) $10a(a - 10b)$

Step1: Distribute $10a$ to both terms

$10a \cdot a - 10a \cdot 10b$

Step2: Simplify each product

$10a^2 - 100ab$

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6) $2(9x - 2y)$

Step1: Distribute $2$ to both terms

$2 \cdot 9x - 2 \cdot 2y$

Step2: Simplify each product

$18x - 4y$

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7) $7x(6x + 4y)$

Step1: Distribute $7x$ to both terms

$7x \cdot 6x + 7x \cdot 4y$

Step2: Simplify each product

$42x^2 + 28xy$

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8) $4a(8a - 8b)$

Step1: Distribute $4a$ to both terms

$4a \cdot 8a - 4a \cdot 8b$

Step2: Simplify each product

$32a^2 - 32ab$

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9) $3n(n^2 - 6n + 5)$

Step1: Distribute $3n$ to all terms

$3n \cdot n^2 - 3n \cdot 6n + 3n \cdot 5$

Step2: Simplify each product

$3n^3 - 18n^2 + 15n$

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10) $2k^3(2k^2 + 5k - 4)$

Step1: Distribute $2k^3$ to all terms

$2k^3 \cdot 2k^2 + 2k^3 \cdot 5k - 2k^3 \cdot 4$

Step2: Simplify each product

$4k^5 + 10k^4 - 8k^3$

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11) $8r^2(4r^2 - 5r + 7)$

Step1: Distribute $8r^2$ to all terms

$8r^2 \cdot 4r^2 - 8r^2 \cdot 5r + 8r^2 \cdot 7$

Step2: Simplify each product

$32r^4 - 40r^3 + 56r^2$

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12) $3(3v^2 + 8v - 5)$

Step1: Distribute $3$ to all terms

$3 \cdot 3v^2 + 3 \cdot 8v - 3 \cdot 5$

Step2: Simplify each product

$9v^2 + 24v - 15$

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13) $7(6x^2 + 9xy + 10y^2)$

Step1: Distribute $7$ to all terms

$7 \cdot 6x^2 + 7 \cdot 9xy + 7 \cdot 10y^2$

Step2: Simplify each product

$42x^2 + 63xy + 70y^2$

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14) $2u(6u^2 - 9uv + v^2)$

Step1: Distribute $2u$ to all terms

$2u \cdot 6u^2 - 2u \cdot 9uv + 2u \cdot v^2$

Step2: Simplify each product

$12u^3 - 18u^2v + 2uv^2$

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15) $9(x^2 + xy - 8y^2)$

Step1: Distribute $9$ to all terms

$9 \cdot x^2 + 9 \cdot xy - 9 \cdot 8y^2$

Step2: Simplify each product

$9x^2 + 9xy - 72y^2$

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16) $9v^2(u^2 + uv - 5v^2)$

Step1: Distribute $9v^2$ to all terms

$9v^2 \cdot u^2 + 9v^2 \cdot uv - 9v^2 \cdot 5v^2$

Step2: Simplify each product

$9u^2v^2 + 9uv^3 - 45v^4$

Answer:

1. $48x^2 + 48x$
2. $42n^2 + 21n$
3. $21r^2 - 24r$
4. $64k - 64$
5. $10a^2 - 100ab$
6. $18x - 4y$
7. $42x^2 + 28xy$
8. $32a^2 - 32ab$
9. $3n^3 - 18n^2 + 15n$
10. $4k^5 + 10k^4 - 8k^3$
11. $32r^4 - 40r^3 + 56r^2$
12. $9v^2 + 24v - 15$
13. $42x^2 + 63xy + 70y^2$
14. $12u^3 - 18u^2v + 2uv^2$
15. $9x^2 + 9xy - 72y^2$
16. $9u^2v^2 + 9uv^3 - 45v^4$