Question
Factor completely. Select "Prime" if the polynomial cannot be factored. \( 1-x^{4}=\square \)
Ask by Thornton Weber. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(1 - x^{4}\) factors to \((1 - x)(1 + x)(1 + x^{2})\).
Solution
Factor the expression \( 1-x^{4} \).
Factor the expression by following steps:
- step0: Factor:
\(1-x^{4}\)
- step1: Rewrite the expression:
\(1^{2}-\left(x^{2}\right)^{2}\)
- step2: Factor the expression:
\(\left(1-x^{2}\right)\left(1+x^{2}\right)\)
- step3: Evaluate:
\(\left(1-x\right)\left(1+x\right)\left(1+x^{2}\right)\)
The factored form of \(1-x^{4}\) is \((1-x)(1+x)(1+x^{2})\).
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Simplify this solution Beyond the Answer
The expression \( 1 - x^{4} \) can be rewritten as a difference of squares: \( (1)^2 - (x^{2})^2 \). This allows us to factor it as \( (1 - x^{2})(1 + x^{2}) \). Next, we can further factor \( 1 - x^{2} \) since it is also a difference of squares: \( (1 - x)(1 + x) \). Therefore, the complete factorization of \( 1 - x^{4} \) is \( (1 - x)(1 + x)(1 + x^{2}) \). So, \( 1 - x^{4} = (1 - x)(1 + x)(1 + x^{2}) \).
