What is factor r^9-8(r^3-1)^3 ?

The solution to factor r^9-8(r^3-1)^3 is -(r^3-2)(7r^6-10r^3+4)

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factor r^9-8(r^3-1)^3

factor

r^{9} - 8 (r^{3} - 1)^{3}

- (r^{3} - 2) (7 r^{6} - 10 r^{3} + 4)

Solution steps

r^{9} - 8 (r^{3} - 1)^{3}

Rewrite

r^{9} - 8 (r^{3} - 1)^{3}

(r^{3})^{3} - (2 (r^{3} - 1))^{3}

= (r^{3})^{3} - (2 (r^{3} - 1))^{3}

Apply Difference of Cubes Formula:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

(r^{3})^{3} - (2 (r^{3} - 1))^{3} = (r^{3} - 2 (r^{3} - 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 2^{2} (r^{3} - 1)^{2})

= (r^{3} - 2 (r^{3} - 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 2^{2} (r^{3} - 1)^{2})

Refine

= (r^{3} - 2 (r^{3} - 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 4 (r^{3} - 1)^{2})

2 (r^{3} - 1) = 2 (r - 1) (r^{2} + r + 1)

= (r^{3} - 2 (r - 1) (r^{2} + r + 1)) (r^{6} + 2 r^{3} (r^{3} - 1) + 4 (r^{3} - 1)^{2})

2 r^{3} (r^{3} - 1) = 2 r^{3} (r - 1) (r^{2} + r + 1)

4 (r^{3} - 1)^{2} = 4 (r - 1)^{2} (r^{2} + r + 1)^{2}

= (r^{3} - 2 (r - 1) (r^{2} + r + 1)) (r^{6} + 2 r^{3} (r - 1) (r^{2} + r + 1) + 4 (r - 1)^{2} (r^{2} + r + 1)^{2})

Factor

r^{3} - 2 (r - 1) (r^{2} + r + 1) : - (r^{3} - 2)

= - (r^{3} - 2) (r^{6} + 2 r^{3} (r - 1) (r^{2} + r + 1) + 4 (r - 1)^{2} (r^{2} + r + 1)^{2})

Expand

r^{6} + 2 r^{3} (r - 1) (r^{2} + r + 1) + 4 (r - 1)^{2} (r^{2} + r + 1)^{2} : 7 r^{6} - 10 r^{3} + 4

= - (r^{3} - 2) (7 r^{6} - 10 r^{3} + 4)

2 x^{2} + 4 = 22

\frac{15}{2}

5 x + 3 < 18

\frac{1}{3} x + \frac{2}{5} x - \frac{1}{2} (x - 3) \leq \frac{1}{10}

6 x^{2} - 19 x + 10 = 0

What is factor r^9-8(r^3-1)^3 ?
The solution to factor r^9-8(r^3-1)^3 is -(r^3-2)(7r^6-10r^3+4)