Question
Which expression is equivalent to \( 7 x^{6}-14 x^{2} \), where \( x>0 \) ? (A) \( 7\left(x^{3}-2 x^{2}\right)^{2} \) (B) \( x^{2}(7 x-2)^{4} \) (C) \( 7 x^{2}(x-2)^{4} \) (D) \( 7 x^{2}\left(x^{4}-2\right) \)
Ask by Hodges Hampton. in the United Arab Emirates
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equivalent expression is \(7x^{2}(x^{4}-2)\), which is option (D).
Solution
1. Start with the expression:
\[
7x^6 - 14x^2
\]
2. Notice that both terms have a common factor \(7x^2\). Factor this out:
\[
7x^6 - 14x^2 = 7x^2\left(x^4 - 2\right)
\]
3. The expression \(7x^2 \left(x^4 - 2\right)\) matches option (D).
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Simplify this solution The Deep Dive
To factor the expression \( 7 x^{6}-14 x^{2} \), we can start by factoring out the common term, which is \( 7 x^{2} \). This gives us: \[ 7 x^{2}(x^{4}-2) \] Now, we can explore the options: - Option (D) gives us \( 7 x^{2}(x^{4}-2) \), which matches perfectly with our factored expression. - Option (A) can be transformed into a different form but won't match the structure we found. - Option (B) has a different factorization than what we've derived. - Option (C) has \( x-2 \) raised to a power, which doesn't align with \( x^{4}-2 \). Thus, the expression equivalent to \( 7 x^{6}-14 x^{2} \) is: \(\boxed{D}\) \( 7 x^{2}\left(x^{4}-2\right) \)
