Polynomial Long Division Calculator

The calculator will perform the long division of polynomials, with steps shown.

• In general, you can skip the multiplication sign, so 5x is equivalent to 5*x.
• In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x).
• Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)).
• If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
• Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x).
• Similarly, tanxsec^3x will be parsed as tan(xsec^3(x)). To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x).
• From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x).
• If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
• All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
 Type Get Constants e e pi pi i i (imaginary unit) Operations a+b a+b a-b a-b a*b a*b a^b, a**b a^b sqrt(x), x^(1/2) sqrt(x) cbrt(x), x^(1/3) root(3)(x) root(x,n), x^(1/n) root(n)(x) x^(a/b) x^(a/b) abs(x) |x| Functions e^x e^x ln(x), log(x) ln(x) ln(x)/ln(a) log_a(x) Trigonometric Functions sin(x) sin(x) cos(x) cos(x) tan(x) tan(x), tg(x) cot(x) cot(x), ctg(x) sec(x) sec(x) csc(x) csc(x), cosec(x) Inverse Trigonometric Functions asin(x), arcsin(x), sin^-1(x) asin(x) acos(x), arccos(x), cos^-1(x) acos(x) atan(x), arctan(x), tan^-1(x) atan(x) acot(x), arccot(x), cot^-1(x) acot(x) asec(x), arcsec(x), sec^-1(x) asec(x) acsc(x), arccsc(x), csc^-1(x) acsc(x) Hyperbolic Functions sinh(x) sinh(x) cosh(x) cosh(x) tanh(x) tanh(x) coth(x) coth(x) 1/cosh(x) sech(x) 1/sinh(x) csch(x) Inverse Hyperbolic Functions asinh(x), arcsinh(x), sinh^-1(x) asinh(x) acosh(x), arccosh(x), cosh^-1(x) acosh(x) atanh(x), arctanh(x), tanh^-1(x) atanh(x) acoth(x), arccoth(x), cot^-1(x) acoth(x) acosh(1/x) asech(x) asinh(1/x) acsch(x)

Divide (dividend):

By (divisor):

Write all suggestions in comments below.

Solution

Your input: find $$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7}$$$using long division. Write the problem in a special format: $$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{x}-7}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrr}\phantom{x^{2}}&\phantom{- 5 x}&\phantom{+3}&\phantom{-17}\end{array}&\\x-7&\phantom{-}\enclose{longdiv}{\begin{array}{cccc}x^{3}&- 12 x^{2}&+38 x&-17\end{array}}&\\\phantom{\color{Magenta}{x}-7}&\begin{array}{rrrr}\end{array}&\begin{array}{c}\end{array}\end{array}$$$

Step 1

Divide the leading term of the dividend by the leading term of the divisor: $$\frac{x^{3}}{x}=x^{2}$$$. Write down the calculated result in the upper part of the table. Multiply it by the divisor: $$x^{2}\left(x-7\right)=x^{3}- 7 x^{2}$$$.

Subtract the dividend from the obtained result: $$\left(x^{3}- 12 x^{2}+38 x-17\right)-\left(x^{3}- 7 x^{2}\right)=- 5 x^{2}+38 x-17$$$. $$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{x}-7}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrr}\color{Fuchsia}{x^{2}}&\phantom{- 5 x}&\phantom{+3}&\phantom{-17}\end{array}&\\\color{Magenta}{x}-7&\phantom{-}\enclose{longdiv}{\begin{array}{cccc}\color{Fuchsia}{x^{3}}&- 12 x^{2}&+38 x&-17\end{array}}&\frac{\color{Fuchsia}{x^{3}}}{\color{Magenta}{x}}=\color{Fuchsia}{x^{2}}\\\phantom{\color{Magenta}{x}-7}&\begin{array}{rrrr}-\phantom{x^{3}}&\phantom{- 12 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}x^{3}&- 7 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&- 5 x^{2}&+38 x&-17\end{array}&\begin{array}{c}\phantom{x^{3}- 12 x^{2}+38 x-17}\\\color{Fuchsia}{x^{2}}\left(\color{Magenta}{x}-7\right)=x^{3}- 7 x^{2}\\\phantom{- 5 x^{2}+38 x-17}\end{array}\end{array}$$$

Step 2

Divide the leading term of the obtained remainder by the leading term of the divisor: $$\frac{- 5 x^{2}}{x}=- 5 x$$$. Write down the calculated result in the upper part of the table. Multiply it by the divisor: $$- 5 x\left(x-7\right)=- 5 x^{2}+35 x$$$.

Subtract the remainder from the obtained result: $$\left(- 5 x^{2}+38 x-17\right)-\left(- 5 x^{2}+35 x\right)=3 x-17$$$. $$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{x}-7}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrr}x^{2}&\color{DeepPink}{- 5 x}&\phantom{+3}&\phantom{-17}\end{array}&\\\color{Magenta}{x}-7&\phantom{-}\enclose{longdiv}{\begin{array}{cccc}x^{3}&- 12 x^{2}&+38 x&-17\end{array}}&\\\phantom{\color{Magenta}{x}-7}&\begin{array}{rrrr}-\phantom{x^{3}}&\phantom{- 12 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}x^{3}&- 7 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&\color{DeepPink}{- 5 x^{2}}&+38 x&-17\\&-\phantom{- 5 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}&- 5 x^{2}&+35 x\\\hline\phantom{\enclose{longdiv}{}}&&3 x&-17\end{array}&\begin{array}{c}\phantom{x^{3}- 12 x^{2}+38 x-17}\\\phantom{\color{Fuchsia}{x^{2}}\left(\color{Magenta}{x}-7\right)=x^{3}- 7 x^{2}}\\\frac{\color{DeepPink}{- 5 x^{2}}}{\color{Magenta}{x}}=\color{DeepPink}{- 5 x}\\\phantom{- 5 x^{2}+38 x-17}\\\color{DeepPink}{- 5 x}\left(\color{Magenta}{x}-7\right)=- 5 x^{2}+35 x\\\phantom{3 x-17}\end{array}\end{array}$$$

Step 3

Divide the leading term of the obtained remainder by the leading term of the divisor: $$\frac{3 x}{x}=3$$$. Write down the calculated result in the upper part of the table. Multiply it by the divisor: $$3\left(x-7\right)=3 x-21$$$.

Subtract the remainder from the obtained result: $$\left(3 x-17\right)-\left(3 x-21\right)=4$$$. $$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{x}-7}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrr}x^{2}&- 5 x&\color{Green}{+3}&\phantom{-17}\end{array}&\\\color{Magenta}{x}-7&\phantom{-}\enclose{longdiv}{\begin{array}{cccc}x^{3}&- 12 x^{2}&+38 x&-17\end{array}}&\\\phantom{\color{Magenta}{x}-7}&\begin{array}{rrrr}-\phantom{x^{3}}&\phantom{- 12 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}x^{3}&- 7 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&- 5 x^{2}&+38 x&-17\\&-\phantom{- 5 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}&- 5 x^{2}&+35 x\\\hline\phantom{\enclose{longdiv}{}}&&\color{Green}{3 x}&-17\\&&-\phantom{3 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}&&3 x&-21\\\hline\phantom{\enclose{longdiv}{}}&&&\color{Crimson}{4}\end{array}&\begin{array}{c}\phantom{x^{3}- 12 x^{2}+38 x-17}\\\phantom{\color{Fuchsia}{x^{2}}\left(\color{Magenta}{x}-7\right)=x^{3}- 7 x^{2}}\\\phantom{- 5 x^{2}+38 x-17}\\\phantom{- 5 x^{2}+38 x-17}\\\phantom{\color{DeepPink}{- 5 x}\left(\color{Magenta}{x}-7\right)=- 5 x^{2}+35 x}\\\frac{\color{Green}{3 x}}{\color{Magenta}{x}}=\color{Green}{3}\\\phantom{3 x-17}\\\color{Green}{3}\left(\color{Magenta}{x}-7\right)=3 x-21\\\phantom{4}\end{array}\end{array}$$$

Since the degree of the remainder is less than the degree of the divisor, then we are done.

The resulting table is shown once more:

$$\require{enclose}\begin{array}{rlc}\phantom{\color{Magenta}{x}-7}&\phantom{\enclose{longdiv}{}-}\begin{array}{rrrr}\color{Fuchsia}{x^{2}}&\color{DeepPink}{- 5 x}&\color{Green}{+3}&\phantom{-17}\end{array}&Hints\\\color{Magenta}{x}-7&\phantom{-}\enclose{longdiv}{\begin{array}{cccc}\color{Fuchsia}{x^{3}}&- 12 x^{2}&+38 x&-17\end{array}}&\frac{\color{Fuchsia}{x^{3}}}{\color{Magenta}{x}}=\color{Fuchsia}{x^{2}}\\\phantom{\color{Magenta}{x}-7}&\begin{array}{rrrr}-\phantom{x^{3}}&\phantom{- 12 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}x^{3}&- 7 x^{2}\\\hline\phantom{\enclose{longdiv}{}}&\color{DeepPink}{- 5 x^{2}}&+38 x&-17\\&-\phantom{- 5 x^{2}}&\phantom{+38 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}&- 5 x^{2}&+35 x\\\hline\phantom{\enclose{longdiv}{}}&&\color{Green}{3 x}&-17\\&&-\phantom{3 x}&\phantom{-17}\\\phantom{\enclose{longdiv}{}}&&3 x&-21\\\hline\phantom{\enclose{longdiv}{}}&&&\color{Crimson}{4}\end{array}&\begin{array}{c}\phantom{x^{3}- 12 x^{2}+38 x-17}\\\color{Fuchsia}{x^{2}}\left(\color{Magenta}{x}-7\right)=x^{3}- 7 x^{2}\\\frac{\color{DeepPink}{- 5 x^{2}}}{\color{Magenta}{x}}=\color{DeepPink}{- 5 x}\\\phantom{- 5 x^{2}+38 x-17}\\\color{DeepPink}{- 5 x}\left(\color{Magenta}{x}-7\right)=- 5 x^{2}+35 x\\\frac{\color{Green}{3 x}}{\color{Magenta}{x}}=\color{Green}{3}\\\phantom{3 x-17}\\\color{Green}{3}\left(\color{Magenta}{x}-7\right)=3 x-21\\\phantom{4}\end{array}\end{array}$$$Therefore, $$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7}=x^{2} - 5 x + 3+\frac{4}{x - 7}$$$

Answer: $$\frac{x^{3} - 12 x^{2} + 38 x - 17}{x - 7}=x^{2} - 5 x + 3+\frac{4}{x - 7}$$\$