Question
Factor. \( a^{3}-a d^{2}-49 a^{2}+49 d^{2} \) \( \begin{array}{l}a^{3}-a d^{2}-49 a^{2}+49 d^{2}= \\ (\text { Factor completely.) }\end{array} \)
Ask by Fuentes Schofield. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (a - 49)(a - d)(a + d) \)
Solution
We start with the expression:
a³ - a·d² - 49a² + 49d²
Step 1. Group the terms:
(a³ - 49a²) + ( - a·d² + 49d²)
Step 2. Factor by grouping:
a²(a - 49) - d²(a - 49)
Step 3. Factor out the common factor (a - 49):
(a - 49)(a² - d²)
Step 4. Notice that a² - d² is a difference of squares:
a² - d² = (a - d)(a + d)
Final Answer:
(a - 49)(a - d)(a + d)
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Simplify this solution Extra Insights
To factor the expression \( a^{3}-a d^{2}-49 a^{2}+49 d^{2} \), we can rewrite it as \( a^{3} - 49a^{2} - ad^{2} + 49d^{2} \). First, we can group the terms: \( (a^{3}-49a^{2}) + (-ad^{2}+49d^{2}) \). Factoring out common factors from each group gives: \( a^{2}(a - 49) - d^{2}(a - 49) \). Now we can factor out \( (a - 49) \): \( (a - 49)(a^{2} - d^{2}) \). Next, we recognize that \( a^{2} - d^{2} \) is a difference of squares, which factors further: \( a^{2} - d^{2} = (a - d)(a + d) \). So the completely factored form of the expression is: \( (a - 49)(a - d)(a + d) \).
