What is factor (x^2-3x+2)^3-216 ?

The solution to factor (x^2-3x+2)^3-216 is (x+1)(x-4)(x^4-6x^3+19x^2-30x+52)

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factor (x^2-3x+2)^3-216

factor

(x^{2} - 3 x + 2)^{3} - 216

(x + 1) (x - 4) (x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 52)

Solution steps

(x^{2} - 3 x + 2)^{3} - 216

Rewrite

216

6^{3}

= (x^{2} - 3 x + 2)^{3} - 6^{3}

Apply Difference of Cubes Formula:

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

(x^{2} - 3 x + 2)^{3} - 6^{3} = ((x^{2} - 3 x + 2) - 6) ((x^{2} - 3 x + 2)^{2} + 6 (x^{2} - 3 x + 2) + 6^{2})

= ((x^{2} - 3 x + 2) - 6) ((x^{2} - 3 x + 2)^{2} + 6 (x^{2} - 3 x + 2) + 6^{2})

Refine

= (x^{2} - 3 x - 4) ((x^{2} - 3 x + 2)^{2} + 6 (x^{2} - 3 x + 2) + 36)

Factor

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

= (x + 1) (x - 4) ((x^{2} - 3 x + 2)^{2} + 6 (x^{2} - 3 x + 2) + 36)

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

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

= (x + 1) (x - 4) ((x - 1)^{2} (x - 2)^{2} + 6 (x - 1) (x - 2) + 36)

Expand

(x - 1)^{2} (x - 2)^{2} + 6 (x - 1) (x - 2) + 36 : x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 52

= (x + 1) (x - 4) (x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 52)

+ a x - 2 b x + a y - 2 b y

- 4 Δ^{2} - 12Δ - 8

5^{2} + 10 x - 15

e^{- t^{2}} y - y

l [t sinh (t)]

What is factor (x^2-3x+2)^3-216 ?
The solution to factor (x^2-3x+2)^3-216 is (x+1)(x-4)(x^4-6x^3+19x^2-30x+52)