What is factor t^{6a}-(t^{2a}-1)^3 ?

The solution to factor t^{6a}-(t^{2a}-1)^3 is 3t^{4a}-3t^{2a}+1

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factor t^{6a}-(t^{2a}-1)^3

factor

t^{6 a} - (t^{2 a} - 1)^{3}

3 t^{4 a} - 3 t^{2 a} + 1

Solution steps

t^{6 a} - (t^{2 a} - 1)^{3}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

t^{6 a} = (t^{a})^{6}

= (t^{a})^{6} - (t^{2 a} - 1)^{3}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

(t^{a})^{6} = ((t^{a})^{2})^{3}

= ((t^{a})^{2})^{3} - (t^{2 a} - 1)^{3}

Apply Difference of Cubes Formula:

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

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

= ((t^{a})^{2} - (t^{2 a} - 1)) ((t^{a})^{4} + (t^{a})^{2} (t^{2 a} - 1) + (t^{2 a} - 1)^{2})

Simplify

((t^{a})^{2} - (t^{2 a} - 1)) ((t^{a})^{4} + (t^{a})^{2} (t^{2 a} - 1) + (t^{2 a} - 1)^{2}) : t^{4 a} + t^{2 a} (t^{2 a} - 1) + (t^{2 a} - 1)^{2}

= t^{4 a} + t^{2 a} (t^{2 a} - 1) + (t^{2 a} - 1)^{2}

Factor

t^{2 a} - 1 : (t^{a} + 1) (t^{a} - 1)

= t^{4 a} + t^{2 a} (t^{a} + 1) (t^{a} - 1) + (t^{2 a} - 1)^{2}

Factor

t^{2 a} - 1 : (t^{a} + 1) (t^{a} - 1)

= t^{4 a} + t^{2 a} (t^{a} + 1) (t^{a} - 1) + ((t^{a} + 1) (t^{a} - 1))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= t^{4 a} + t^{2 a} (t^{a} + 1) (t^{a} - 1) + (t^{a} + 1)^{2} (t^{a} - 1)^{2}

(t^{a} + 1)^{2} : t^{2 a} + 2 t^{a} + 1

= t^{4 a} + t^{2 a} (t^{a} + 1) (t^{a} - 1) + (t^{2 a} + 2 t^{a} + 1) (t^{a} - 1)^{2}

(t^{a} - 1)^{2} : t^{2 a} - 2 t^{a} + 1

= t^{4 a} + t^{2 a} (t^{a} + 1) (t^{a} - 1) + (t^{2 a} + 2 t^{a} + 1) (t^{2 a} - 2 t^{a} + 1)

Expand

t^{2 a} (t^{a} + 1) (t^{a} - 1) : t^{4 a} - t^{2 a}

= t^{4 a} + t^{4 a} - t^{2 a} + (t^{2 a} + 2 t^{a} + 1) (t^{2 a} - 2 t^{a} + 1)

Expand

(t^{2 a} + 2 t^{a} + 1) (t^{2 a} - 2 t^{a} + 1) : - 2 t^{2 a} + t^{4 a} + 1

= t^{4 a} + t^{4 a} - t^{2 a} - 2 t^{2 a} + t^{4 a} + 1

Simplify

t^{4 a} + t^{4 a} - t^{2 a} - 2 t^{2 a} + t^{4 a} + 1 : 3 t^{4 a} - 3 t^{2 a} + 1

= 3 t^{4 a} - 3 t^{2 a} + 1

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What is factor t^{6a}-(t^{2a}-1)^3 ?
The solution to factor t^{6a}-(t^{2a}-1)^3 is 3t^{4a}-3t^{2a}+1