Question

divide.

1. $(x^3 + 15x^2 + 47x - 33) \div (x + 10)$
2. $(5y^3 - 25y^2 - 75y + 22) \div (y - 7)$
3. $(12x^2 + 30x - 19) \div (4x - 2)$
4. $(5x^3 - 33x^2 - 12x + 20) \div (5x - 3)$
5. $(m^3 - 2m^2 + 7) \div (m - 2)$
6. $(x^4 - 9x^3 + 1) \div (x - 9)$

Explanation:

Problem 11: $(x^3 + 15x^2 + 47x - 33) \div (x + 10)$

Step1: Divide leading terms

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

Step2: Multiply divisor by $x^2$

$x^2(x+10) = x^3 + 10x^2$

Step3: Subtract from dividend

$(x^3+15x^2+47x-33)-(x^3+10x^2) = 5x^2 + 47x$

Step4: Divide new leading terms

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

Step5: Multiply divisor by $5x$

$5x(x+10) = 5x^2 + 50x$

Step6: Subtract from new polynomial

$(5x^2+47x)-(5x^2+50x) = -3x - 33$

Step7: Divide new leading terms

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

Step8: Multiply divisor by $-3$

$-3(x+10) = -3x - 30$

Step9: Subtract to get remainder

$(-3x-33)-(-3x-30) = -3$

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Problem 12: $(5v^3 -25v^2 -73v +22) \div (v - 7)$

Step1: Divide leading terms

$\frac{5v^3}{v} = 5v^2$

Step2: Multiply divisor by $5v^2$

$5v^2(v-7) = 5v^3 -35v^2$

Step3: Subtract from dividend

$(5v^3-25v^2-73v+22)-(5v^3-35v^2) = 10v^2 -73v$

Step4: Divide new leading terms

$\frac{10v^2}{v} = 10v$

Step5: Multiply divisor by $10v$

$10v(v-7) = 10v^2 -70v$

Step6: Subtract from new polynomial

$(10v^2-73v)-(10v^2-70v) = -3v +22$

Step7: Divide new leading terms

$\frac{-3v}{v} = -3$

Step8: Multiply divisor by $-3$

$-3(v-7) = -3v +21$

Step9: Subtract to get remainder

$(-3v+22)-(-3v+21) = 1$

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Problem 13: $(12x^2 +30x -19) \div (4x - 2)$

Step1: Divide leading terms

$\frac{12x^2}{4x} = 3x$

Step2: Multiply divisor by $3x$

$3x(4x-2) = 12x^2 -6x$

Step3: Subtract from dividend

$(12x^2+30x-19)-(12x^2-6x) = 36x -19$

Step4: Divide new leading terms

$\frac{36x}{4x} = 9$

Step5: Multiply divisor by $9$

$9(4x-2) = 36x -18$

Step6: Subtract to get remainder

$(36x-19)-(36x-18) = -1$

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Problem 14: $(5x^3 -33x^2 -12x +20) \div (5x - 3)$

Step1: Divide leading terms

$\frac{5x^3}{5x} = x^2$

Step2: Multiply divisor by $x^2$

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

Step3: Subtract from dividend

$(5x^3-33x^2-12x+20)-(5x^3-3x^2) = -30x^2 -12x$

Step4: Divide new leading terms

$\frac{-30x^2}{5x} = -6x$

Step5: Multiply divisor by $-6x$

$-6x(5x-3) = -30x^2 +18x$

Step6: Subtract from new polynomial

$(-30x^2-12x)-(-30x^2+18x) = -30x +20$

Step7: Divide new leading terms

$\frac{-30x}{5x} = -6$

Step8: Multiply divisor by $-6$

$-6(5x-3) = -30x +18$

Step9: Subtract to get remainder

$(-30x+20)-(-30x+18) = 2$

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Problem 15: $(m^3 -2m^2 +7) \div (m - 2)$

Step1: Divide leading terms

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

Step2: Multiply divisor by $m^2$

$m^2(m-2) = m^3 -2m^2$

Step3: Subtract from dividend

$(m^3-2m^2+7)-(m^3-2m^2) = 7$

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Problem 16: $(x^4 -9x^3 +1) \div (x - 9)$

Step1: Divide leading terms

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

Step2: Multiply divisor by $x^3$

$x^3(x-9) = x^4 -9x^3$

Step3: Subtract from dividend

$(x^4-9x^3+1)-(x^4-9x^3) = 1$

Answer:

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