Question
Factor the difference of cubes. Select "Prime" if the polynomial cannot be factored.
Ask by George Mckenzie. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
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.
Solution
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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Simplify this solution 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!
