Use long division method to solve x³−19 / x+4

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to solve the division using the long division method:

We need to divide $x^3 - y^3$ by $x+y$ . To make the long division clearer, we can write the dividend as $x^3 + 0x^2y + 0xy^2 - y^3$ .

Divide the first term of the dividend ( $x^3$ ) by the first term of the divisor ( $x$ ).
$\frac{x^3}{x} = x^2$
Write $x^2$ as the first term of the quotient.
Multiply the quotient term ( $x^2$ ) by the entire divisor ( $x+y$ ).
$x^2(x+y) = x^3 + x^2y$
Subtract this result from the dividend.
$(x^3 + 0x^2y + 0xy^2 - y^3) - (x^3 + x^2y) = -x^2y + 0xy^2 - y^3$
Bring down the next term ( $0xy^2$ ) and repeat the process.
Divide the first term of the new polynomial ( $-x^2y$ ) by the first term of the divisor ( $x$ ).
$\frac{-x^2y}{x} = -xy$
Write $-xy$ as the next term of the quotient.
Multiply the new quotient term ( $-xy$ ) by the entire divisor ( $x+y$ ).
$-xy(x+y) = -x^2y - xy^2$
Subtract this result from the current polynomial.
$(-x^2y + 0xy^2 - y^3) - (-x^2y - xy^2) = xy^2 - y^3$
Bring down the next term ( $-y^3$ ) and repeat the process.
Divide the first term of the new polynomial ( $xy^2$ ) by the first term of the divisor ( $x$ ).
$\frac{xy^2}{x} = y^2$
Write $y^2$ as the next term of the quotient.
Multiply the new quotient term ( $y^2$ ) by the entire divisor ( $x+y$ ).
$y^2(x+y) = xy^2 + y^3$
Subtract this result from the current polynomial.
$(xy^2 - y^3) - (xy^2 + y^3) = -2y^3$
This is the remainder.

The long division can be visualized as:
$\begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -xy & +y^2 \\ \cline{2-5} x+y & x^3 & +0x^2y & +0xy^2 & -y^3 \\ \multicolumn{2}{r}{-(x^3} & +x^2y) \\ \cline{2-3} \multicolumn{2}{r}{0} & -x^2y & +0xy^2 \\ \multicolumn{2}{r}{} & -(-x^2y & -xy^2) \\ \cline{3-4} \multicolumn{2}{r}{} & 0 & xy^2 & -y^3 \\ \multicolumn{2}{r}{} & & -(xy^2 & +y^3) \\ \cline{4-5} \multicolumn{2}{r}{} & & 0 & -2y^3 \\ \end{array}$
The quotient is $x^2 - xy + y^2$ and the remainder is $-2y^3$ .
Therefore, the result of the division is:
$\frac{x^3 - y^3}{x+y} = \boxed*{x^2 - xy + y^2 - \frac{2y^3}{x+y}}$

Use long division method to solve x³-19 / x+4

Step-by-step solution

Divide the first term of the dividend ( $x^3$ ) by the first term of the divisor ( $x$ ).

Multiply the quotient term ( $x^2$ ) by the entire divisor ( $x+y$ ).

Subtract this result from the dividend.

Bring down the next term ( $0xy^2$ ) and repeat the process.

Multiply the new quotient term ( $-xy$ ) by the entire divisor ( $x+y$ ).

Subtract this result from the current polynomial.

Bring down the next term ( $-y^3$ ) and repeat the process.

Multiply the new quotient term ( $y^2$ ) by the entire divisor ( $x+y$ ).

Subtract this result from the current polynomial.

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More Mathematics Questions

Use long division method to solve x³-19 / x+4

Step-by-step solution

Divide the first term of the dividend (x3x^3x3) by the first term of the divisor (xxx).

Multiply the quotient term (x2x^2x2) by the entire divisor (x+yx+yx+y).

Subtract this result from the dividend.

Bring down the next term (0xy20xy^20xy2) and repeat the process.

Multiply the new quotient term (−xy-xy−xy) by the entire divisor (x+yx+yx+y).

Subtract this result from the current polynomial.

Bring down the next term (−y3-y^3−y3) and repeat the process.

Multiply the new quotient term (y2y^2y2) by the entire divisor (x+yx+yx+y).

Subtract this result from the current polynomial.

Still stuck on this one?

More Mathematics Questions

Divide the first term of the dividend ( $x^3$ ) by the first term of the divisor ( $x$ ).

Multiply the quotient term ( $x^2$ ) by the entire divisor ( $x+y$ ).

Bring down the next term ( $0xy^2$ ) and repeat the process.

Multiply the new quotient term ( $-xy$ ) by the entire divisor ( $x+y$ ).

Bring down the next term ( $-y^3$ ) and repeat the process.

Multiply the new quotient term ( $y^2$ ) by the entire divisor ( $x+y$ ).