Question
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Answer
\( x^{4}-256 = (x-4)(x+4)(x^{2}+16) \)
Solution
Factor the expression \( x^4-256 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{4}-256\)
- step1: Rewrite the expression:
\(\left(x^{2}\right)^{2}-\left(256^{\frac{1}{2}}\right)^{2}\)
- step2: Factor the expression:
\(\left(x^{2}-256^{\frac{1}{2}}\right)\left(x^{2}+256^{\frac{1}{2}}\right)\)
- step3: Evaluate:
\(\left(x^{2}-16\right)\left(x^{2}+256^{\frac{1}{2}}\right)\)
- step4: Evaluate:
\(\left(x^{2}-16\right)\left(x^{2}+16\right)\)
- step5: Evaluate:
\(\left(x-4\right)\left(x+4\right)\left(x^{2}+16\right)\)
The factored form of \( x^{4}-256 \) is \((x-4)(x+4)(x^{2}+16)\).
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Simplify this solution Beyond the Answer
To factor \( x^{4} - 256 \), we can recognize that it is a difference of squares. We can express it as \( (x^{2})^{2} - (16)^{2} \), which factors into \( (x^{2} - 16)(x^{2} + 16) \). Now, we can further factor \( x^{2} - 16 \) since it is also a difference of squares: \( x^{2} - 16 = (x - 4)(x + 4) \). The term \( x^{2} + 16 \) does not factor over the real numbers but can be factored over the complex numbers as \( (x - 4i)(x + 4i) \), where \( i \) is the imaginary unit. So, the complete factorization of \( x^{4} - 256 \) is: \[ (x - 4)(x + 4)(x^{2} + 16) \] or, if factoring over the complex numbers, \[ (x - 4)(x + 4)(x - 4i)(x + 4i). \]
