Question
\( 1 \leftarrow \quad \begin{array}{l}\text { Factor } \\ \quad b^{2}+12 b f+32 f^{2}\end{array} \)
Ask by Ramirez Wells. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The factored form is \((b+8f)(b+4f)\).
Solution
Factor the expression \( b^{2}+12b*f+32f^{2} \).
Factor the expression by following steps:
- step0: Factor:
\(b^{2}+12bf+32f^{2}\)
- step1: Rewrite the expression:
\(b^{2}+\left(4+8\right)bf+32f^{2}\)
- step2: Calculate:
\(b^{2}+4bf+8fb+32f^{2}\)
- step3: Rewrite the expression:
\(b\times b+b\times 4f+8fb+8f\times 4f\)
- step4: Factor the expression:
\(b\left(b+4f\right)+8f\left(b+4f\right)\)
- step5: Factor the expression:
\(\left(b+8f\right)\left(b+4f\right)\)
The factored form of the expression \(b^{2}+12bf+32f^{2}\) is \((b+8f)(b+4f)\).
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Simplify this solution Extra Insights
To factor the quadratic expression \( b^{2}+12bf+32f^{2} \), we are looking for two numbers that multiply to \( 32f^2 \) and add up to \( 12f \). The numbers \( 4f \) and \( 8f \) work perfectly since \( 4f + 8f = 12f \) and \( 4f \times 8f = 32f^2 \). Thus, we can rewrite the expression as: \[ b^{2} + 4bf + 8bf + 32f^{2} = (b + 4f)(b + 8f) \] So, the fully factored form of the expression is \( (b + 4f)(b + 8f) \). For a quick recap, always look for pairs of numbers that match your needed conditions—this is a key skill in factoring. And remember to check your work by expanding your factors to ensure you return to the original expression!
