factorize x^3-27 — step-by-step solution

1) Let's factorize in

ℝ

x^{3} - 27

⇥1.1) The roots of

x^{3} - 27

are the solutions of

x^{3} = 27

⇥The complex solutions are:

\sqrt[3]{27} \cdot \exp (i \cdot \frac{2 k π}{3})

, with k ranging from 0 to 2

⇥

k	Exponential solution	Transformed and simplified as
0	$3 \exp (i \cdot \frac{2 \cdot 0 \cdot π}{3}) = 3$	$3 \cdot (\cos (0) + i \cdot \sin (0)) = 3$
1	$3 \exp (i \cdot \frac{2 \cdot 1 \cdot π}{3}) = 3 \exp (\frac{i \cdot 2 π}{3})$	$3 \cdot (\cos (\frac{2 π}{3}) + i \cdot \sin (\frac{2 π}{3})) = \frac{3 \cdot (- 1 + i \cdot \sqrt{3})}{2}$
2	$3 \exp (i \cdot \frac{2 \cdot 2 \cdot π}{3}) = 3 \exp (\frac{i \cdot 4 π}{3})$	$3 \cdot (\cos (\frac{4 π}{3}) + i \cdot \sin (\frac{4 π}{3})) = - \frac{3 \cdot (1 + i \cdot \sqrt{3})}{2}$

⇥For the real solutions, factorization gives:

x - 3

⇥1.2) Compute the quotient and remainder of

(x^{3} - 27)

(x - 3)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3} - 27$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 3) = x^{3} - 3 x^{2}$	$(x^{3} - 27) - (x^{3} - 3 x^{2}) = 3 x^{2} - 27$
$3 x^{2} - 27$	$\frac{3 x^{2}}{x}$	$3 x$	$3 x \cdot (x - 3) = 3 x^{2} - 9 x$	$(3 x^{2} - 27) - (3 x^{2} - 9 x) = 9 x - 27$
$9 x - 27$	$\frac{9 x}{x}$	$9$	$9 \cdot (x - 3) = 9 x - 27$	$(9 x - 27) - (9 x - 27) = 0$

⇥So the result of the Euclidean division is:

x^{2} + 3 x + 9

ⓘ The other roots,

- \frac{3 \cdot (1 + i \cdot \sqrt{3})}{2}

\frac{3 \cdot (- 1 + i \cdot \sqrt{3})}{2}

, are not real:

x^{2} + 3 x + 9

cannot be factored in ℝ

▶Factors found:

(x - 3) \cdot (x^{2} + 3 x + 9)