Pregunta
Find the product. \( \frac{x+5}{x^{3}-1} \cdot\left(x^{2}+x+1\right)=\square \)
Ask by Tyler Reeves. in the United States
Mar 12,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The product is \( \frac{x+5}{x-1} \).
Solución
Calculate or simplify the expression \( (x+5)/(x^3-1) * (x^2+x+1) \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{\left(x+5\right)}{\left(x^{3}-1\right)}\times \left(x^{2}+x+1\right)\)
- step1: Remove the parentheses:
\(\frac{x+5}{x^{3}-1}\times \left(x^{2}+x+1\right)\)
- step2: Rewrite the expression:
\(\frac{x+5}{\left(x^{2}+x+1\right)\left(x-1\right)}\times \left(x^{2}+x+1\right)\)
- step3: Reduce the fraction:
\(\frac{x+5}{x-1}\times 1\)
- step4: Multiply the terms:
\(\frac{x+5}{x-1}\)
The product of \( \frac{x+5}{x^{3}-1} \) and \( x^{2}+x+1 \) is \( \frac{x+5}{x-1} \).
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Simplifique esta solución Extra Insights
To find the product \( \frac{x+5}{x^{3}-1} \cdot\left(x^{2}+x+1\right) \), we first note that \( x^3 - 1 \) can be factored using the identity for the difference of cubes: \[ x^3 - 1 = (x - 1)(x^2 + x + 1) \] Now, substituting this factorization into the expression, we have: \[ \frac{x+5}{(x - 1)(x^2 + x + 1)} \cdot (x^2 + x + 1) \] The \( x^2 + x + 1 \) terms will cancel out, assuming \( x^2 + x + 1 \neq 0 \) (which is valid for \( x \neq 1 \)). So, the expression simplifies to: \[ \frac{x+5}{x-1} \] Thus, the final answer is: \[ \frac{x+5}{x-1} \]
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