Pregunta
1)) Factor \( x^{4}+4 x^{2}-5 \) completely. 1)) All factors in your answer should have integer coefficients.
Ask by Gross Morgan. in the United States
Mar 12,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( x^{4}+4x^{2}-5 = (x-1)(x+1)(x^{2}+5) \)
Solución
Factor the expression \( x^4+4x^2-5 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{4}+4x^{2}-5\)
- step1: Rewrite the expression:
\(x^{4}+\left(5-1\right)x^{2}-5\)
- step2: Calculate:
\(x^{4}+5x^{2}-x^{2}-5\)
- step3: Rewrite the expression:
\(x^{2}\times x^{2}+x^{2}\times 5-x^{2}-5\)
- step4: Factor the expression:
\(x^{2}\left(x^{2}+5\right)-\left(x^{2}+5\right)\)
- step5: Factor the expression:
\(\left(x^{2}-1\right)\left(x^{2}+5\right)\)
- step6: Factor the expression:
\(\left(x-1\right)\left(x+1\right)\left(x^{2}+5\right)\)
The complete factorization of \(x^{4}+4x^{2}-5\) is \((x-1)(x+1)(x^{2}+5)\).
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Simplifique esta solución Beyond the Answer
To factor \( x^{4} + 4x^{2} - 5 \) completely, we can start by using a substitution. Let \( y = x^{2} \), transforming the expression into \( y^{2} + 4y - 5 \). Now, we factor the quadratic: \[ y^{2} + 4y - 5 = (y + 5)(y - 1) \] Substituting back \( y = x^{2} \), we get: \[ (x^{2} + 5)(x^{2} - 1) \] The term \( x^{2} - 1 \) can be further factored using the difference of squares: \[ x^{2} - 1 = (x + 1)(x - 1) \] Putting it all together, the complete factorization of \( x^{4} + 4x^{2} - 5 \) is: \[ (x^{2} + 5)(x + 1)(x - 1) \]
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