What is factor n^{10}+n^8+n^6-n^4-n^2-1 ?

The solution to factor n^{10}+n^8+n^6-n^4-n^2-1 is (n^2+1+n)^2(n^2+1-n)^2(n+1)(n-1)

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factor n^{10}+n^8+n^6-n^4-n^2-1

factor

n^{10} + n^{8} + n^{6} - n^{4} - n^{2} - 1

(n^{2} + 1 + n)^{2} (n^{2} + 1 - n)^{2} (n + 1) (n - 1)

Solution steps

n^{10} + n^{8} + n^{6} - n^{4} - n^{2} - 1

Let

u = n^{2}

= u^{5} + u^{4} + u^{3} - u^{2} - u - 1

Factor

u^{5} + u^{4} + u^{3} - u^{2} - u - 1 : (u^{2} + u + 1) (u - 1) (u^{2} + u + 1)

= (u^{2} + u + 1) (u - 1) (u^{2} + u + 1)

Substitute back

u = n^{2}

= (n^{4} + n^{2} + 1) (n^{2} - 1) (n^{4} + n^{2} + 1)

Factor

n^{4} + n^{2} + 1 : (n^{2} + 1 + n) (n^{2} + 1 - n)

= (n^{2} + 1 + n) (n^{2} + 1 - n) (n^{2} - 1) (n^{4} + n^{2} + 1)

Factor

n^{2} - 1 : (n + 1) (n - 1)

= (n^{2} + 1 + n) (n^{2} + 1 - n) (n + 1) (n - 1) (n^{4} + n^{2} + 1)

Factor

n^{4} + n^{2} + 1 : (n^{2} + 1 + n) (n^{2} + 1 - n)

= (n^{2} + 1 + n) (n^{2} + 1 - n) (n + 1) (n - 1) (n^{2} + 1 + n) (n^{2} + 1 - n)

Simplify

= (n^{2} + 1 + n)^{2} (n^{2} + 1 - n)^{2} (n + 1) (n - 1)

\frac{2 x}{x - 2} - \frac{x + 3}{x + 5}

2 \frac{3}{4} + 1 \frac{2}{5}

\frac{24 x ^{7} y ^{7}}{40 x y ^{6}}

4 x^{3} + 2 x - 2

y^{36}

What is factor n^{10}+n^8+n^6-n^4-n^2-1 ?
The solution to factor n^{10}+n^8+n^6-n^4-n^2-1 is (n^2+1+n)^2(n^2+1-n)^2(n+1)(n-1)