Question

find each quotient using long division.

1. $(k^3 - 10k^2 + k + 1)\div(k - 1)$
2. $(x^4 + 4x^3 - 28x^2 - 45x + 26)\div(x + 7)$
3. $(20c^3 + 22c^2 - 7c + 7)\div(5c - 2)$
4. $(3n^4 + 6n^3 - 15n^2 + 32n - 25)\div(n + 4)$

Explanation:

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Problem 1: $(k^3 - 10k^2 + k + 1) \div (k - 1)$

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

$(k^3 -10k^2 +k +1) - (k^3 -k^2) = -9k^2 +k +1$

Step4: Divide new leading term

$\frac{-9k^2}{k} = -9k$

Step5: Multiply divisor by $-9k$

$-9k(k-1) = -9k^2 +9k$

Step6: Subtract from new polynomial

$(-9k^2 +k +1) - (-9k^2 +9k) = -8k +1$

Step7: Divide new leading term

$\frac{-8k}{k} = -8$

Step8: Multiply divisor by $-8$

$-8(k-1) = -8k +8$

Step9: Subtract to get remainder

$(-8k +1) - (-8k +8) = -7$

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Problem 2: $(x^4 + 4x^3 -28x^2 -45x +26) \div (x + 7)$

Step1: Divide leading terms

$\frac{x^4}{x} = x^3$

Step2: Multiply divisor by $x^3$

$x^3(x+7) = x^4 +7x^3$

Step3: Subtract from dividend

$(x^4 +4x^3 -28x^2 -45x +26) - (x^4 +7x^3) = -3x^3 -28x^2 -45x +26$

Step4: Divide new leading term

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

Step5: Multiply divisor by $-3x^2$

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

Step6: Subtract from new polynomial

$(-3x^3 -28x^2 -45x +26) - (-3x^3 -21x^2) = -7x^2 -45x +26$

Step7: Divide new leading term

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

Step8: Multiply divisor by $-7x$

$-7x(x+7) = -7x^2 -49x$

Step9: Subtract from new polynomial

$(-7x^2 -45x +26) - (-7x^2 -49x) = 4x +26$

Step10: Divide new leading term

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

Step11: Multiply divisor by $4$

$4(x+7) = 4x +28$

Step12: Subtract to get remainder

$(4x +26) - (4x +28) = -2$

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Problem 3: $(20c^3 +22c^2 -7c +7) \div (5c -2)$

Step1: Divide leading terms

$\frac{20c^3}{5c} = 4c^2$

Step2: Multiply divisor by $4c^2$

$4c^2(5c-2) = 20c^3 -8c^2$

Step3: Subtract from dividend

$(20c^3 +22c^2 -7c +7) - (20c^3 -8c^2) = 30c^2 -7c +7$

Step4: Divide new leading term

$\frac{30c^2}{5c} = 6c$

Step5: Multiply divisor by $6c$

$6c(5c-2) = 30c^2 -12c$

Step6: Subtract from new polynomial

$(30c^2 -7c +7) - (30c^2 -12c) = 5c +7$

Step7: Divide new leading term

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

Step8: Multiply divisor by $1$

$1(5c-2) = 5c -2$

Step9: Subtract to get remainder

$(5c +7) - (5c -2) = 9$

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Problem 4: $(3n^4 +6n^3 -15n^2 +32n -25) \div (n + 4)$

Step1: Divide leading terms

$\frac{3n^4}{n} = 3n^3$

Step2: Multiply divisor by $3n^3$

$3n^3(n+4) = 3n^4 +12n^3$

Step3: Subtract from dividend

$(3n^4 +6n^3 -15n^2 +32n -25) - (3n^4 +12n^3) = -6n^3 -15n^2 +32n -25$

Step4: Divide new leading term

$\frac{-6n^3}{n} = -6n^2$

Step5: Multiply divisor by $-6n^2$

$-6n^2(n+4) = -6n^3 -24n^2$

Step6: Subtract from new polynomial

$(-6n^3 -15n^2 +32n -25) - (-6n^3 -24n^2) = 9n^2 +32n -25$

Step7: Divide new leading term

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

Step8: Multiply divisor by $9n$

$9n(n+4) = 9n^2 +36n$

Step9: Subtract from new polynomial

$(9n^2 +32n -25) - (9n^2 +36n) = -4n -25$

Step10: Divide new leading term

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

Step11: Multiply divisor by $-4$

$-4(n+4) = -4n -16$

Step12: Subtract to get remainder

$(-4n -25) - (-4n -16) = -9$

Answer:

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