Question
6. \( 27 a^{3}-3 a b^{2} \)
Ask by Olson West. in South Africa
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \( 27a^{3} - 3ab^{2} \) can be factored into \( 3a(3a - b)(3a + b) \).
Solution
Factor the expression by following steps:
- step0: Factor:
\(27a^{3}-3ab^{2}\)
- step1: Factor the expression:
\(3a\left(9a^{2}-b^{2}\right)\)
- step2: Factor the expression:
\(3a\left(3a-b\right)\left(3a+b\right)\)
The expression \( 27 a^{3} - 3 a b^{2} \) can be factored as follows:
1. **Identify the common factor**: The common factor in both terms is \( 3a \).
2. **Factor out the common factor**: When we factor \( 3a \) out of the expression, we get:
\[
3a(9a^{2} - b^{2})
\]
3. **Recognize the difference of squares**: The expression \( 9a^{2} - b^{2} \) is a difference of squares, which can be factored further:
\[
9a^{2} - b^{2} = (3a - b)(3a + b)
\]
Putting it all together, the fully factored form of the expression is:
\[
3a(3a - b)(3a + b)
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Mind Expander
To factor the expression \( 27 a^{3}-3 a b^{2} \), start by identifying the greatest common factor (GCF). In this case, the GCF is \( 3a \). Factoring out \( 3a \) gives us: \[ 3a(9a^{2} - b^{2}) \] Now, you have a difference of squares in the parentheses. The difference of squares can be factored further: \[ 9a^{2} - b^{2} = (3a - b)(3a + b) \] Thus, the completely factored form of the expression is: \[ 3a(3a - b)(3a + b) \] And there you have it! A clean factored expression for \( 27 a^{3}-3 a b^{2} \). Happy factoring!
