What is expand (x-1/x)^6 ?

The solution to expand (x-1/x)^6 is x^6-6x^4+15x^2-20+(15)/(x^2)-6/(x^4)+1/(x^6)

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expand (x-1/x)^6

expand

(x - \frac{1}{x})^{6}

x^{6} - 6 x^{4} + 15 x^{2} - 20 + \frac{15}{x ^{2}} - \frac{6}{x ^{4}} + \frac{1}{x ^{6}}

Solution steps

(x - \frac{1}{x})^{6}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = - \frac{1}{x}

= i = 0 \sum 6 (i 6) x^{(6 - i)} (- \frac{1}{x})^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (- \frac{1}{x})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (- \frac{1}{x})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (- \frac{1}{x})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (- \frac{1}{x})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (- \frac{1}{x})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (- \frac{1}{x})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (- \frac{1}{x})^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (- \frac{1}{x})^{0} : x^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (- \frac{1}{x})^{1} : - 6 x^{4}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (- \frac{1}{x})^{2} : 15 x^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (- \frac{1}{x})^{3} : - 20

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (- \frac{1}{x})^{4} : \frac{15}{x ^{2}}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (- \frac{1}{x})^{5} : - \frac{6}{x ^{4}}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (- \frac{1}{x})^{6} : \frac{1}{x ^{6}}

= x^{6} - 6 x^{4} + 15 x^{2} - 20 + \frac{15}{x ^{2}} - \frac{6}{x ^{4}} + \frac{1}{x ^{6}}

(ln (x))^{2}

(ln (x) + y^{3}) d x - 3 x y^{2} d y

M (x, y) d x + (e^{15 x} - 3 x^{2} y) d y

\frac{π ( 3 ln ( 3 ) - 2 )}{4}

(5 + x)^{4} (1 + \frac{1}{x ^{2}})

What is expand (x-1/x)^6 ?
The solution to expand (x-1/x)^6 is x^6-6x^4+15x^2-20+(15)/(x^2)-6/(x^4)+1/(x^6)