Question
Which expression is a factor of both \( x^{2}-9 \) and \( x^{2}+8 x+15 ? \) \( (x+5) \)
Ask by Hodgson Collins. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The common factor is \( (x + 3) \).
Solution
To determine a common factor, factor both expressions.
1. Factor x² – 9:
x² – 9 = (x – 3)(x + 3)
2. Factor x² + 8x + 15:
x² + 8x + 15 = (x + 3)(x + 5)
The common factor between the two is (x + 3).
Therefore, the expression (x + 3) is a factor of both, not (x + 5).
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Simplify this solution The Deep Dive
To find a common factor of \( x^{2}-9 \) and \( x^{2}+8x+15 \), we can factor both expressions. The expression \( x^{2}-9 \) is a difference of squares, which factors to \( (x-3)(x+3) \). The expression \( x^{2}+8x+15 \) factors to \( (x+3)(x+5) \). Noticing that both factorizations share the common factor \( (x+3) \), we conclude that \( (x+3) \) is indeed a factor of both expressions. To check your understanding of factoring, remember that identifying the greatest common factor can simplify problem-solving significantly. A common mistake is assuming that expressions without obvious common terms don’t have shared factors—always explore the algebra!
