Question

1. $(x^2 - 74) div (x - 8)$
2. $(2p^2 + 7p - 39) div (2p - 7)$
3. $(n^3 + 7n^2 + 14n + 3) div (n + 2)$
4. $(p^3 - 10p^2 + 20p + 26) div (p - 5)$
5. $(v^3 - 2v^2 - 14v - 5) div (v + 3)$
6. $(x^3 - 13x^2 + 40x + 18) div (x - 7)$
7. $(k^3 - 30k - 18 - 4k^2) div (3 + k)$
8. $(-5k^2 + k^3 + 8k + 4) div (-1 + k)$
9. $(x^3 + 5x^2 - 32x - 7) div (x - 4)$
10. $(50k^3 + 10k^2 - 35k - 7) div (5k - 4)$

Explanation:

11) $(x^2 - 74) \div (x - 8)$

Step1: Divide leading terms

$\frac{x^2}{x} = x$

Step2: Multiply divisor by $x$

$x(x-8) = x^2 - 8x$

Step3: Subtract from dividend

$(x^2 - 74) - (x^2 - 8x) = 8x - 74$

Step4: Divide new leading term

$\frac{8x}{x} = 8$

Step5: Multiply divisor by 8

$8(x-8) = 8x - 64$

Step6: Subtract to get remainder

$(8x - 74) - (8x - 64) = -10$

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12) $(2p^2 + 7p - 39) \div (2p - 7)$

Step1: Divide leading terms

$\frac{2p^2}{2p} = p$

Step2: Multiply divisor by $p$

$p(2p-7) = 2p^2 - 7p$

Step3: Subtract from dividend

$(2p^2 + 7p - 39) - (2p^2 - 7p) = 14p - 39$

Step4: Divide new leading term

$\frac{14p}{2p} = 7$

Step5: Multiply divisor by 7

$7(2p-7) = 14p - 49$

Step6: Subtract to get remainder

$(14p - 39) - (14p - 49) = 10$

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13) $(n^3 + 7n^2 + 14n + 3) \div (n + 2)$

Step1: Divide leading terms

$\frac{n^3}{n} = n^2$

Step2: Multiply divisor by $n^2$

$n^2(n+2) = n^3 + 2n^2$

Step3: Subtract from dividend

$(n^3 + 7n^2 + 14n + 3) - (n^3 + 2n^2) = 5n^2 + 14n + 3$

Step4: Divide new leading term

$\frac{5n^2}{n} = 5n$

Step5: Multiply divisor by $5n$

$5n(n+2) = 5n^2 + 10n$

Step6: Subtract from new polynomial

$(5n^2 + 14n + 3) - (5n^2 + 10n) = 4n + 3$

Step7: Divide new leading term

$\frac{4n}{n} = 4$

Step8: Multiply divisor by 4

$4(n+2) = 4n + 8$

Step9: Subtract to get remainder

$(4n + 3) - (4n + 8) = -5$

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14) $(p^3 - 10p^2 + 20p + 26) \div (p - 5)$

Step1: Divide leading terms

$\frac{p^3}{p} = p^2$

Step2: Multiply divisor by $p^2$

$p^2(p-5) = p^3 - 5p^2$

Step3: Subtract from dividend

$(p^3 - 10p^2 + 20p + 26) - (p^3 - 5p^2) = -5p^2 + 20p + 26$

Step4: Divide new leading term

$\frac{-5p^2}{p} = -5p$

Step5: Multiply divisor by $-5p$

$-5p(p-5) = -5p^2 + 25p$

Step6: Subtract from new polynomial

$(-5p^2 + 20p + 26) - (-5p^2 + 25p) = -5p + 26$

Step7: Divide new leading term

$\frac{-5p}{p} = -5$

Step8: Multiply divisor by $-5$

$-5(p-5) = -5p + 25$

Step9: Subtract to get remainder

$(-5p + 26) - (-5p + 25) = 1$

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15) $(v^3 - 2v^2 - 14v - 5) \div (v + 3)$

Step1: Divide leading terms

$\frac{v^3}{v} = v^2$

Step2: Multiply divisor by $v^2$

$v^2(v+3) = v^3 + 3v^2$

Step3: Subtract from dividend

$(v^3 - 2v^2 - 14v - 5) - (v^3 + 3v^2) = -5v^2 - 14v - 5$

Step4: Divide new leading term

$\frac{-5v^2}{v} = -5v$

Step5: Multiply divisor by $-5v$

$-5v(v+3) = -5v^2 - 15v$

Step6: Subtract from new polynomial

$(-5v^2 - 14v - 5) - (-5v^2 - 15v) = v - 5$

Step7: Divide new leading term

$\frac{v}{v} = 1$

Step8: Multiply divisor by 1

$1(v+3) = v + 3$

Step9: Subtract to get remainder

$(v - 5) - (v + 3) = -8$

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16) $(x^3 - 13x^2 + 40x + 18) \div (x - 7)$

Step1: Divide leading terms

$\frac{x^3}{x} = x^2$

Step2: Multiply divisor by $x^2$

$x^2(x-7) = x^3 - 7x^2$

Step3: Subtract from dividend

$(x^3 - 13x^2 + 40x + 18) - (x^3 - 7x^2) = -6x^2 + 40x + 18$

Step4: Divide new leading term

$\frac{-6x^2}{x} = -6x$

Step5: Multiply divisor by $-6x$

$-6x(x-7) = -6x^2 + 42x$

Step6: Subtract from new polynomial

$(-6x^2 + 40x + 18) - (-6x^2 + 42x) = -2x + 18$

Step7: Divide new leading term

$\frac{-2x}{x} = -2$

Step8: Multiply divisor by $-2$

$-2(x-7) = -2x + 14$

Step9: Subtract to get remainder

$(-2x + 18) - (-2x + 14) = 4$

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17) $(k^3 - 4k^2 - 30k - 18) \div (k + 3)$ (rearranged dividend)

Step1: Divide leading terms

$\frac{k^3}{k} = k^2$

Step2: Multiply divisor by $k^2$

$k^2(k+3) = k^3 + 3k^2$

Step3: Subtract from dividend

$(k^3 - 4k^2 - 30k - 18) - (k^3 + 3k^2) = -7k^2 - 30k - 18$

Step4: Divide new leading term

$\frac{-7k^2}{k} = -7k$

Step5: Multiply divisor by $-7k$

$-7k(k+3) = -7k^2 - 21k$

Step6: Subtract from new polynomial

$(-7k^2 - 30k - 18) - (-7k^2 - 21k) = -9k - 18$

Step7: Divide new leading term

$\frac{-9k}{k} = -9$

Step8: Multiply divisor by $-9$

$-9(k+3) = -9k - 27$

Step9: Subtract to get remainder

$(-9k - 18) - (-9k - 27) = 9$

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18) $(k^3 - 5k^2 + 8k + 4) \div (k - 1)$ (rearranged dividend)

Step1: Divide leading terms

$\frac{k^3}{k} = k^2$

Step2: Multiply divisor by $k^2$

$k^2(k-1) = k^3 - k^2$

Step3: Subtract from dividend

$(k^3 - 5k^2 + 8k + 4) - (k^3 - k^2) = -4k^2 + 8k + 4$

Step4: Divide new leading term

$\frac{-4k^2}{k} = -4k$

Step5: Multiply divisor by $-4k$

$-4k(k-1) = -4k^2 + 4k$

Step6: Subtract from new polynomial

$(-4k^2 + 8k + 4) - (-4k^2 + 4k) = 4k + 4$

Step7: Divide new leading term

$\frac{4k}{k} = 4$

Step8: Multiply divisor by 4

$4(k-1) = 4k - 4$

Step9: Subtract to get remainder

$(4k + 4) - (4k - 4) = 8$

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19) $(x^3 + 5x^2 - 32x - 7) \div (x - 4)$

Step1: Divide leading terms

$\frac{x^3}{x} = x^2$

Step2: Multiply divisor by $x^2$

$x^2(x-4) = x^3 - 4x^2$

Step3: Subtract from dividend

$(x^3 + 5x^2 - 32x - 7) - (x^3 - 4x^2) = 9x^2 - 32x - 7$

Step4: Divide new leading term

$\frac{9x^2}{x} = 9x$

Step5: Multiply divisor by $9x$

$9x(x-4) = 9x^2 - 36x$

Step6: Subtract from new polynomial

$(9x^2 - 32x - 7) - (9x^2 - 36x) = 4x - 7$

Step7: Divide new leading term

$\frac{4x}{x} = 4$

Step8: Multiply divisor by 4

$4(x-4) = 4x - 16$

Step9: Subtract to get remainder

$(4x - 7) - (4x - 16) = 9$

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20) $(50k^3 + 10k^2 - 35k - 7) \div (5k - 4)$

Step1: Divide leading terms

$\frac{50k^3}{5k} = 10k^2$

Step2: Multiply divisor by $10k^2$

$10k^2(5k-4) = 50k^3 - 40k^2$

Step3: Subtract from dividend

$(50k^3 + 10k^2 - 35k - 7) - (50k^3 - 40k^2) = 50k^2 - 35k - 7$

Step4: Divide new leading term

$\frac{50k^2}{5k} = 10k$

Step5: Multiply divisor by $10k$

$10k(5k-4) = 50k^2 - 40k$

Step6: Subtract from new polynomial

$(50k^2 - 35k - 7) - (50k^2 - 40k) = 5k - 7$

Step7: Divide new leading term

$\frac{5k}{5k} = 1$

Step8: Multiply divisor by 1

$1(5k-4) = 5k - 4$

Step9: Subtract to get remainder

$(5k - 7) - (5k - 4) = -3$

Answer:

1. $x + 8 + \frac{-10}{x-8}$
2. $p + 7 + \frac{10}{2p-7}$
3. $n^2 + 5n + 4 + \frac{-5}{n+2}$
4. $p^2 - 5p - 5 + \frac{1}{p-5}$
5. $v^2 - 5v + 1 + \frac{-8}{v+3}$
6. $x^2 - 6x - 2 + \frac{4}{x-7}$
7. $k^2 - 7k - 9 + \frac{9}{k+3}$
8. $k^2 - 4k + 4 + \frac{8}{k-1}$
9. $x^2 + 9x + 4 + \frac{9}{x-4}$
10. $10k^2 + 10k + 1 + \frac{-3}{5k-4}$