factor x^8-2x^4+1 — step-by-step solution

⚠Ⓘ 2x is read as the product 2*x: for a power, write 2^x

1) Let's factor in

ℝ

x^{8} - 2 x^{4} + 1

You can rewrite the expression as:

{(x^{4} - 1)}^{2}

2) Let's factor in

ℝ

x^{4} - 1

The (complex) solutions are:

\exp (i \cdot \frac{2 k π}{4})

for k from 0 to 3

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

In ℝ, factoring gives:

(x + 1) \cdot (x - 1)

x \cdot x

+ x \cdot (-1)

+ 1 \cdot x

+ 1 \cdot (-1)

ℹ

x \cdot x

x^{2}

‖

x \cdot (-1)

- x

x^{2}

- x

+ x

- 1

x^{2} + 0 - 1

x^{2} - 1

3) Let's start the euclidean division of

x^{4} - 1

x^{2} - 1

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{4} - 1$	$\frac{x^{4}}{x^{2}}$	$x^{2}$	$x^{2} \cdot (x^{2} - 1) = x^{4} - x^{2}$	$(x^{4} - 1) - (x^{4} - x^{2}) = x^{2} - 1$
$x^{2} - 1$	$\frac{x^{2}}{x^{2}}$	$1$	$x^{2} - 1 = x^{2} - 1$	$(x^{2} - 1) - (x^{2} - 1) = 0$

So the result of the euclidean division is:

x^{2} + 1

Factors found:

(x^{2} + 1) \cdot (x + 1) \cdot (x - 1)

▶Factored expression:

{(x^{2} + 1)}^{2} \cdot {((x + 1) \cdot (x - 1))}^{2}