factor 16-x^4 — step-by-step solution

1) Let's factorize in

ℝ

16 - x^{4}

Solutions are:

16^{\frac{1}{4}} \cdot \exp (i \cdot \frac{2 k π}{4})

(with k = 0..3)

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

The factorization for the real solution(s) yields:

(x + 2) \cdot (x - 2)

x \cdot x

+ x \cdot (-2)

+ 2 x

+ 2 \cdot (-2)

ℹ

x \cdot x

x^{2}

‖

2 \cdot (-2)

-4

x^{2}

- x \cdot 2

+ 2 x

- 4

x^{2} + 0 - 4

x^{2} - 4

2) Perform polynomial long division of

(- x^{4} + 16)

(x^{2} - 4)

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$- x^{4} + 16$	$- \frac{x^{4}}{x^{2}}$	$- x^{2}$	$- x^{2} \cdot (x^{2} - 4) = - x^{4} + 4 x^{2}$	$(- x^{4} + 16) - (- x^{4} + 4 x^{2}) = - 4 x^{2} + 16$
$- 4 x^{2} + 16$	$- \frac{4 x^{2}}{x^{2}}$	$-4$	$-4 \cdot (x^{2} - 4) = - 4 x^{2} + 16$	$(- 4 x^{2} + 16) - (- 4 x^{2} + 16) = 0$

Therefore, the result of the euclidean division is:

- x^{2} - 4

Factors found:

(- x^{2} - 4) \cdot (x + 2) \cdot (x - 2)

▶Factored expression:

(x + 2) \cdot (x - 2) \cdot (- x^{2} - 4)