Pregunta
Factor the difference of cubes. Select "Prime" if the polynomial cannot be factored.
Ask by George Mckenzie. in the United States
Mar 09,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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To factor a difference of cubes, use the formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
If the polynomial doesn't fit this form, it's prime.
Solución
1. Recognize that a difference of cubes can be written in the form
\[
a^3 - b^3
\]
where \(a\) and \(b\) are expressions.
2. Use the difference of cubes factorization formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2).
\]
3. Substitute the appropriate expressions for \(a\) and \(b\) if given a specific polynomial. Otherwise, the factored form of the difference of cubes remains as above.
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Simplifique esta solución Bonus Knowledge
The difference of cubes can be factored using the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). For example, if you have \(x^3 - 27\), you would identify \(a\) as \(x\) and \(b\) as \(3\) (since \(27 = 3^3\)). Applying the formula: \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\). Easy peasy! When factoring, a common mistake is overlooking the necessity of cubes. If the expression isn't in the form \(a^3 - b^3\), you can’t use this method. Always check the structure first! If it doesn't match, you may just end up with a prime polynomial instead. Stay sharp!
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