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Answer :
To find which expression is equivalent to [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex], let's break down the expression step by step:
1. Look inside the parentheses:
- We have [tex]\( (4x^4 - 3x^2) \)[/tex].
2. Square the expression inside the parentheses:
- [tex]\( (4x^4 - 3x^2)^2 \)[/tex] means multiplying [tex]\( (4x^4 - 3x^2) \)[/tex] by itself:
- [tex]\((4x^4 - 3x^2)(4x^4 - 3x^2) = 16x^8 - 12x^6 - 12x^6 + 9x^4 \)[/tex].
- Simplifying this gives: [tex]\( 16x^8 - 24x^6 + 9x^4 \)[/tex].
3. Multiply by [tex]\(3x\)[/tex]:
- Now, distribute [tex]\(3x\)[/tex] across the squared expression:
- [tex]\(3x(16x^8 - 24x^6 + 9x^4) = 48x^9 - 72x^7 + 27x^5 \)[/tex].
4. Combine with the rest of the expression:
- Add [tex]\(9x^5\)[/tex] to the result:
- [tex]\(9x^5 + 48x^9 - 72x^7 + 27x^5 = 48x^9 - 72x^7 + 36x^5\)[/tex].
The expression simplifies to [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
Comparing this result to the given options, the equivalent expression is:
- [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
So, the correct choice from the options is:
- [tex]\(48 x^9 - 72 x^7 + 36 x^5\)[/tex]
1. Look inside the parentheses:
- We have [tex]\( (4x^4 - 3x^2) \)[/tex].
2. Square the expression inside the parentheses:
- [tex]\( (4x^4 - 3x^2)^2 \)[/tex] means multiplying [tex]\( (4x^4 - 3x^2) \)[/tex] by itself:
- [tex]\((4x^4 - 3x^2)(4x^4 - 3x^2) = 16x^8 - 12x^6 - 12x^6 + 9x^4 \)[/tex].
- Simplifying this gives: [tex]\( 16x^8 - 24x^6 + 9x^4 \)[/tex].
3. Multiply by [tex]\(3x\)[/tex]:
- Now, distribute [tex]\(3x\)[/tex] across the squared expression:
- [tex]\(3x(16x^8 - 24x^6 + 9x^4) = 48x^9 - 72x^7 + 27x^5 \)[/tex].
4. Combine with the rest of the expression:
- Add [tex]\(9x^5\)[/tex] to the result:
- [tex]\(9x^5 + 48x^9 - 72x^7 + 27x^5 = 48x^9 - 72x^7 + 36x^5\)[/tex].
The expression simplifies to [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
Comparing this result to the given options, the equivalent expression is:
- [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
So, the correct choice from the options is:
- [tex]\(48 x^9 - 72 x^7 + 36 x^5\)[/tex]
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