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

The solution to expand (x^6-1)/(x-1) is x^5+x^4+x^3+x^2+x+1

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expand (x^6-1)/(x-1)

expand

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

x^{5} + x^{4} + x^{3} + x^{2} + x + 1

Solution steps

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

Factor

x^{6} - 1 : (x + 1) (x - 1) (x^{4} + x^{2} + 1)

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

Cancel the common factor:

x - 1

= (x + 1) (x^{4} + x^{2} + 1)

Distribute parentheses

= x x^{4} + x x^{2} + x \cdot 1 + 1 \cdot x^{4} + 1 \cdot x^{2} + 1 \cdot 1

= x^{4} x + x^{2} x + 1 \cdot x + 1 \cdot x^{4} + 1 \cdot x^{2} + 1 \cdot 1

Simplify

x^{4} x + x^{2} x + 1 \cdot x + 1 \cdot x^{4} + 1 \cdot x^{2} + 1 \cdot 1 : x^{5} + x^{3} + x + x^{4} + x^{2} + 1

= x^{5} + x^{3} + x + x^{4} + x^{2} + 1

Rewrite in standard form

= x^{5} + x^{4} + x^{3} + x^{2} + x + 1

e^{x} (x y + y^{2} + y)

\frac{1}{( s + 1 ) ^{2} ( s + 2 )}

\frac{d y}{( 2 y + 3 ) ^{2}}

\frac{x + 1}{x}

\frac{1}{s ( s ^{2} + 2 s + 5 )}

What is expand (x^6-1)/(x-1) ?
The solution to expand (x^6-1)/(x-1) is x^5+x^4+x^3+x^2+x+1