What is factor (x+b+1)^6+(z+c-1)^6 ?

The solution to factor (x+b+1)^6+(z+c-1)^6 is

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factor (x+b+1)^6+(z+c-1)^6

factor

(x + b + 1)^{6} + (z + c - 1)^{6}

(x^{2} + 2 b x + 2 x + b^{2} + 2 b + c^{2} + 2 cz + z^{2} + 2 - 2 c - 2 z) (x^{4} + 4 b x^{3} + 4 x^{3} + 6 b^{2} x^{2} + 12 b x^{2} + 2 c x^{2} + 2 x^{2} z + 5 x^{2} - c^{2} x^{2} - x^{2} z^{2} - 2 c x^{2} z + 4 b^{3} x + 12 b^{2} x + 10 b x + 4 b c x + 4 c x + 4 b x z + 4 x z + 2 x - 2 b c^{2} x - 2 c^{2} x - 2 b x z^{2} - 2 x z^{2} - 4 b c x z - 4 c x z + b^{4} + 4 b^{3} + 5 b^{2} + 2 b^{2} c + 2 b^{2} z + 2 b + 4 b c + 4 b z + c^{4} + 4 c^{3} z + 5 c^{2} + 6 c^{2} z^{2} + 4 c z^{3} + 10 cz + z^{4} + 5 z^{2} + 1 - b^{2} c^{2} - b^{2} z^{2} - 2 b^{2} cz - 2 b c^{2} - 2 b z^{2} - 4 b cz - 4 c^{3} - 12 c^{2} z - 2 c - 12 c z^{2} - 4 z^{3} - 2 z)

Solution steps

(x + b + 1)^{6} + (z + c - 1)^{6}

Rewrite

(x + b + 1)^{6} + (z + c - 1)^{6}

((x + b + 1)^{2})^{3} + ((z + c - 1)^{2})^{3}

= ((x + b + 1)^{2})^{3} + ((z + c - 1)^{2})^{3}

Apply Sum of Cubes Formula:

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

((x + b + 1)^{2})^{3} + ((z + c - 1)^{2})^{3} = ((x + b + 1)^{2} + (z + c - 1)^{2}) ((x + b + 1)^{4} - (x + b + 1)^{2} (z + c - 1)^{2} + (z + c - 1)^{4})

= ((x + b + 1)^{2} + (z + c - 1)^{2}) ((x + b + 1)^{4} - (x + b + 1)^{2} (z + c - 1)^{2} + (z + c - 1)^{4})

Expand

(x + b + 1)^{2} + (z + c - 1)^{2} : x^{2} + 2 b x + 2 x + b^{2} + 2 b + c^{2} + 2 cz + z^{2} + 2 - 2 c - 2 z

= (x^{2} + 2 b x + 2 x + b^{2} + 2 b + c^{2} + 2 cz + z^{2} + 2 - 2 c - 2 z) ((x + b + 1)^{4} - (x + b + 1)^{2} (z + c - 1)^{2} + (z + c - 1)^{4})

Expand

(x + b + 1)^{4} - (x + b + 1)^{2} (z + c - 1)^{2} + (z + c - 1)^{4} : x^{4} + 4 b x^{3} + 4 x^{3} + 6 b^{2} x^{2} + 12 b x^{2} + 2 c x^{2} + 2 x^{2} z + 5 x^{2} - c^{2} x^{2} - x^{2} z^{2} - 2 c x^{2} z + 4 b^{3} x + 12 b^{2} x + 10 b x + 4 b c x + 4 c x + 4 b x z + 4 x z + 2 x - 2 b c^{2} x - 2 c^{2} x - 2 b x z^{2} - 2 x z^{2} - 4 b c x z - 4 c x z + b^{4} + 4 b^{3} + 5 b^{2} + 2 b^{2} c + 2 b^{2} z + 2 b + 4 b c + 4 b z + c^{4} + 4 c^{3} z + 5 c^{2} + 6 c^{2} z^{2} + 4 c z^{3} + 10 cz + z^{4} + 5 z^{2} + 1 - b^{2} c^{2} - b^{2} z^{2} - 2 b^{2} cz - 2 b c^{2} - 2 b z^{2} - 4 b cz - 4 c^{3} - 12 c^{2} z - 2 c - 12 c z^{2} - 4 z^{3} - 2 z

= (x^{2} + 2 b x + 2 x + b^{2} + 2 b + c^{2} + 2 cz + z^{2} + 2 - 2 c - 2 z) (x^{4} + 4 b x^{3} + 4 x^{3} + 6 b^{2} x^{2} + 12 b x^{2} + 2 c x^{2} + 2 x^{2} z + 5 x^{2} - c^{2} x^{2} - x^{2} z^{2} - 2 c x^{2} z + 4 b^{3} x + 12 b^{2} x + 10 b x + 4 b c x + 4 c x + 4 b x z + 4 x z + 2 x - 2 b c^{2} x - 2 c^{2} x - 2 b x z^{2} - 2 x z^{2} - 4 b c x z - 4 c x z + b^{4} + 4 b^{3} + 5 b^{2} + 2 b^{2} c + 2 b^{2} z + 2 b + 4 b c + 4 b z + c^{4} + 4 c^{3} z + 5 c^{2} + 6 c^{2} z^{2} + 4 c z^{3} + 10 cz + z^{4} + 5 z^{2} + 1 - b^{2} c^{2} - b^{2} z^{2} - 2 b^{2} cz - 2 b c^{2} - 2 b z^{2} - 4 b cz - 4 c^{3} - 12 c^{2} z - 2 c - 12 c z^{2} - 4 z^{3} - 2 z)

3 + (7)

f (4 t)

1 + 2 cos (2 x) + cos^{2} (2 x)

ln (e^{5} - e^{3})

(2 x - 7)^{2} - 3 (2 x - 7) - 88

What is factor (x+b+1)^6+(z+c-1)^6 ?
The solution to factor (x+b+1)^6+(z+c-1)^6 is