Question

divide the polynomial and simplify:
$x + 1 \overline{) x^2 - 9x - 10}$
$x + 7 \overline{) x^2 + 9x + 14}$
$x^2 + 3x - 5 \overline{) x^4 + 4x^3 + x - 10}$

Explanation:

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First division: $\boldsymbol{\frac{x^2 - 9x - 10}{x+1}}$

Step1: Factor the numerator

$x^2 - 9x - 10 = (x+1)(x-10)$

Step2: Cancel common factors

$\frac{(x+1)(x-10)}{x+1} = x-10$
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Second division: $\boldsymbol{\frac{x^2 + 9x + 14}{x+7}}$

Step1: Factor the numerator

$x^2 + 9x + 14 = (x+7)(x+2)$

Step2: Cancel common factors

$\frac{(x+7)(x+2)}{x+7} = x+2$
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Third division: $\boldsymbol{\frac{x^4 + 4x^3 + x - 10}{x^2 + 3x - 5}}$

Step1: Divide leading terms

$\frac{x^4}{x^2} = x^2$; multiply divisor: $x^2(x^2+3x-5)=x^4+3x^3-5x^2$

Step2: Subtract from dividend

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

Step3: Divide new leading terms

$\frac{x^3}{x^2}=x$; multiply divisor: $x(x^2+3x-5)=x^3+3x^2-5x$

Step4: Subtract from new dividend

$(x^3+5x^2+x-10)-(x^3+3x^2-5x)=2x^2+6x-10$

Step5: Divide new leading terms

$\frac{2x^2}{x^2}=2$; multiply divisor: $2(x^2+3x-5)=2x^2+6x-10$

Step6: Subtract to find remainder

$(2x^2+6x-10)-(2x^2+6x-10)=0$

Step7: Combine quotient terms

$x^2 + x + 2$
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Answer:

1. $x-10$
2. $x+2$
3. $x^2 + x + 2$