Solved on Jan 23, 2024

Find the quotient of $x - 2$ divided by $x^{3} - 5x^{2} - x - 10$ .

STEP 1

Assumptions
1. We are performing polynomial long division.
2. We are dividing the polynomial $x^3 - 5x^2 - x - 10$ by the linear polynomial $x - 2$ .

STEP 2

Set up the long division by writing the divisor $x - 2$ outside the division box and the dividend $x^3 - 5x^2 - x - 10$ inside the division box.

STEP 3

Divide the first term of the dividend, $x^3$ , by the first term of the divisor, $x$ , to get the first term of the quotient.
$\frac{x^3}{x} = x^2$

STEP 4

Multiply the divisor $x - 2$ by the first term of the quotient $x^2$ to subtract from the dividend.
$x^2 \cdot (x - 2) = x^3 - 2x^2$

STEP 5

Subtract the result of the multiplication from the dividend.
\begin{align} &\phantom{-}x^3 - 5x^2 - x - 10\\ -&\underline{x^3 - 2x^2}\\ &\phantom{x^3} -3x^2 - x - 10 \end{align}

STEP 6

Bring down the next term of the dividend to prepare for the next step of division.
$-3x^2 - x$

STEP 7

Divide the first term of the new dividend, $-3x^2$ , by the first term of the divisor, $x$ , to get the next term of the quotient.
$\frac{-3x^2}{x} = -3x$

STEP 8

Multiply the divisor $x - 2$ by the new term of the quotient $-3x$ to subtract from the new dividend.
$-3x \cdot (x - 2) = -3x^2 + 6x$

STEP 9

Subtract the result of the multiplication from the new dividend.
\begin{align} &\phantom{-} -3x^2 - x - 10\\ -&\underline{-3x^2 + 6x}\\ &\phantom{-3x^2} -7x - 10 \end{align}

STEP 10

Bring down the next term of the dividend to prepare for the next step of division.
$-7x - 10$

STEP 11

Divide the first term of the new dividend, $-7x$ , by the first term of the divisor, $x$ , to get the next term of the quotient.
$\frac{-7x}{x} = -7$

STEP 12

Multiply the divisor $x - 2$ by the new term of the quotient $-7$ to subtract from the new dividend.
$-7 \cdot (x - 2) = -7x + 14$

STEP 13

Subtract the result of the multiplication from the new dividend.
\begin{align} &\phantom{-} -7x - 10\\ -&\underline{-7x + 14}\\ &\phantom{-7x} -24 \end{align}

STEP 14

Since the degree of the remainder $-24$ is less than the degree of the divisor $x - 2$ , we cannot continue the division process. Thus, $-24$ is the remainder.

STEP 15

Combine all the terms of the quotient and write down the remainder to express the result of the division.
$x^2 - 3x - 7 - \frac{24}{x - 2}$
The quotient is $x^2 - 3x - 7$ with a remainder of $-24$ .

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Find the quotient of $x - 2$ divided by $x^{3} - 5x^{2} - x - 10$ .

STEP 1

Assumptions
1. We are performing polynomial long division.
2. We are dividing the polynomial $x^3 - 5x^2 - x - 10$ by the linear polynomial $x - 2$ .

STEP 2

Set up the long division by writing the divisor $x - 2$ outside the division box and the dividend $x^3 - 5x^2 - x - 10$ inside the division box.

STEP 3

Divide the first term of the dividend, $x^3$ , by the first term of the divisor, $x$ , to get the first term of the quotient.
$\frac{x^3}{x} = x^2$

STEP 4

Multiply the divisor $x - 2$ by the first term of the quotient $x^2$ to subtract from the dividend.
$x^2 \cdot (x - 2) = x^3 - 2x^2$

STEP 5

Subtract the result of the multiplication from the dividend.
\begin{align} &\phantom{-}x^3 - 5x^2 - x - 10\\ -&\underline{x^3 - 2x^2}\\ &\phantom{x^3} -3x^2 - x - 10 \end{align}

STEP 6

Bring down the next term of the dividend to prepare for the next step of division.
$-3x^2 - x$

STEP 7

Divide the first term of the new dividend, $-3x^2$ , by the first term of the divisor, $x$ , to get the next term of the quotient.
$\frac{-3x^2}{x} = -3x$

STEP 8

Multiply the divisor $x - 2$ by the new term of the quotient $-3x$ to subtract from the new dividend.
$-3x \cdot (x - 2) = -3x^2 + 6x$

STEP 9

Subtract the result of the multiplication from the new dividend.
\begin{align} &\phantom{-} -3x^2 - x - 10\\ -&\underline{-3x^2 + 6x}\\ &\phantom{-3x^2} -7x - 10 \end{align}

STEP 10

Bring down the next term of the dividend to prepare for the next step of division.
$-7x - 10$

STEP 11

Divide the first term of the new dividend, $-7x$ , by the first term of the divisor, $x$ , to get the next term of the quotient.
$\frac{-7x}{x} = -7$

STEP 12

Multiply the divisor $x - 2$ by the new term of the quotient $-7$ to subtract from the new dividend.
$-7 \cdot (x - 2) = -7x + 14$

STEP 13

Subtract the result of the multiplication from the new dividend.
\begin{align} &\phantom{-} -7x - 10\\ -&\underline{-7x + 14}\\ &\phantom{-7x} -24 \end{align}

STEP 14

Since the degree of the remainder $-24$ is less than the degree of the divisor $x - 2$ , we cannot continue the division process. Thus, $-24$ is the remainder.

STEP 15

Combine all the terms of the quotient and write down the remainder to express the result of the division.
$x^2 - 3x - 7 - \frac{24}{x - 2}$
The quotient is $x^2 - 3x - 7$ with a remainder of $-24$ .

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Find the quotient of x−2x - 2x−2 divided by x3−5x2−x−10x^{3} - 5x^{2} - x - 10x3−5x2−x−10.

STEP 1

Assumptions1. We are performing polynomial long division.2. We are dividing the polynomial x3−5x2−x−10x^3 - 5x^2 - x - 10x3−5x2−x−10 by the linear polynomial x−2x - 2x−2.

STEP 2

Set up the long division by writing the divisor x−2x - 2x−2 outside the division box and the dividend x3−5x2−x−10x^3 - 5x^2 - x - 10x3−5x2−x−10 inside the division box.

STEP 3

Divide the first term of the dividend, x3x^3x3, by the first term of the divisor, xxx, to get the first term of the quotient.x3x=x2\frac{x^3}{x} = x^2xx3​=x2

STEP 4

Multiply the divisor x−2x - 2x−2 by the first term of the quotient x2x^2x2 to subtract from the dividend.x2⋅(x−2)=x3−2x2x^2 \cdot (x - 2) = x^3 - 2x^2x2⋅(x−2)=x3−2x2

STEP 5

Subtract the result of the multiplication from the dividend.\begin{align*} &\phantom{-}x^3 - 5x^2 - x - 10\\ -&\underline{x^3 - 2x^2}\\ &\phantom{x^3} -3x^2 - x - 10 \end{align*}

STEP 6

Bring down the next term of the dividend to prepare for the next step of division.−3x2−x-3x^2 - x−3x2−x

STEP 7

Divide the first term of the new dividend, −3x2-3x^2−3x2, by the first term of the divisor, xxx, to get the next term of the quotient.−3x2x=−3x\frac{-3x^2}{x} = -3xx−3x2​=−3x

STEP 8

Multiply the divisor x−2x - 2x−2 by the new term of the quotient −3x-3x−3x to subtract from the new dividend.−3x⋅(x−2)=−3x2+6x-3x \cdot (x - 2) = -3x^2 + 6x−3x⋅(x−2)=−3x2+6x

STEP 9

Subtract the result of the multiplication from the new dividend.\begin{align*} &\phantom{-} -3x^2 - x - 10\\ -&\underline{-3x^2 + 6x}\\ &\phantom{-3x^2} -7x - 10 \end{align*}

STEP 10

Bring down the next term of the dividend to prepare for the next step of division.−7x−10-7x - 10−7x−10

STEP 11

Divide the first term of the new dividend, −7x-7x−7x, by the first term of the divisor, xxx, to get the next term of the quotient.−7xx=−7\frac{-7x}{x} = -7x−7x​=−7

STEP 12

Multiply the divisor x−2x - 2x−2 by the new term of the quotient −7-7−7 to subtract from the new dividend.−7⋅(x−2)=−7x+14-7 \cdot (x - 2) = -7x + 14−7⋅(x−2)=−7x+14

STEP 13

Subtract the result of the multiplication from the new dividend.\begin{align*} &\phantom{-} -7x - 10\\ -&\underline{-7x + 14}\\ &\phantom{-7x} -24 \end{align*}

STEP 14

Since the degree of the remainder −24-24−24 is less than the degree of the divisor x−2x - 2x−2, we cannot continue the division process. Thus, −24-24−24 is the remainder.

STEP 15

Combine all the terms of the quotient and write down the remainder to express the result of the division.x2−3x−7−24x−2x^2 - 3x - 7 - \frac{24}{x - 2}x2−3x−7−x−224​The quotient is x2−3x−7x^2 - 3x - 7x2−3x−7 with a remainder of −24-24−24.

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Find the quotient of $x - 2$ divided by $x^{3} - 5x^{2} - x - 10$ .

Assumptions
1. We are performing polynomial long division.
2. We are dividing the polynomial $x^3 - 5x^2 - x - 10$ by the linear polynomial $x - 2$ .

Set up the long division by writing the divisor $x - 2$ outside the division box and the dividend $x^3 - 5x^2 - x - 10$ inside the division box.

Divide the first term of the dividend, $x^3$ , by the first term of the divisor, $x$ , to get the first term of the quotient.
$\frac{x^3}{x} = x^2$

Multiply the divisor $x - 2$ by the first term of the quotient $x^2$ to subtract from the dividend.
$x^2 \cdot (x - 2) = x^3 - 2x^2$

Subtract the result of the multiplication from the dividend.
\begin{align} &\phantom{-}x^3 - 5x^2 - x - 10\\ -&\underline{x^3 - 2x^2}\\ &\phantom{x^3} -3x^2 - x - 10 \end{align}

Bring down the next term of the dividend to prepare for the next step of division.
$-3x^2 - x$

Divide the first term of the new dividend, $-3x^2$ , by the first term of the divisor, $x$ , to get the next term of the quotient.
$\frac{-3x^2}{x} = -3x$

Multiply the divisor $x - 2$ by the new term of the quotient $-3x$ to subtract from the new dividend.
$-3x \cdot (x - 2) = -3x^2 + 6x$

Subtract the result of the multiplication from the new dividend.
\begin{align} &\phantom{-} -3x^2 - x - 10\\ -&\underline{-3x^2 + 6x}\\ &\phantom{-3x^2} -7x - 10 \end{align}

Bring down the next term of the dividend to prepare for the next step of division.
$-7x - 10$

Divide the first term of the new dividend, $-7x$ , by the first term of the divisor, $x$ , to get the next term of the quotient.
$\frac{-7x}{x} = -7$

Multiply the divisor $x - 2$ by the new term of the quotient $-7$ to subtract from the new dividend.
$-7 \cdot (x - 2) = -7x + 14$

Subtract the result of the multiplication from the new dividend.
\begin{align} &\phantom{-} -7x - 10\\ -&\underline{-7x + 14}\\ &\phantom{-7x} -24 \end{align}

Since the degree of the remainder $-24$ is less than the degree of the divisor $x - 2$ , we cannot continue the division process. Thus, $-24$ is the remainder.

Combine all the terms of the quotient and write down the remainder to express the result of the division.
$x^2 - 3x - 7 - \frac{24}{x - 2}$
The quotient is $x^2 - 3x - 7$ with a remainder of $-24$ .