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Answer :
To find an equivalent expression for [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex], we can start by simplifying the expression inside the parentheses and then multiplying it out.
1. Simplify [tex]\( (4x^4 - 3x^2)^2 \)[/tex]:
Begin by expanding the square:
- [tex]\((4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2\)[/tex].
- Calculating each term gives [tex]\(16x^8 - 24x^6 + 9x^4\)[/tex].
2. Multiply the simplified expression by [tex]\(3x\)[/tex]:
Distribute [tex]\(3x\)[/tex] across each term:
- [tex]\(3x \cdot 16x^8 = 48x^9\)[/tex],
- [tex]\(3x \cdot (-24x^6) = -72x^7\)[/tex],
- [tex]\(3x \cdot 9x^4 = 27x^5\)[/tex].
3. Combine these with the original [tex]\(9x^5\)[/tex]:
Add the polynomial from the previous step to [tex]\(9x^5\)[/tex]:
- The terms are [tex]\(48x^9 - 72x^7 + 27x^5 + 9x^5\)[/tex].
- Combine like terms: [tex]\(27x^5 + 9x^5 = 36x^5\)[/tex].
So, the expression [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex] simplifies to [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
Thus, the correct equivalent expression is:
[tex]\[48x^9 - 72x^7 + 36x^5\][/tex]
1. Simplify [tex]\( (4x^4 - 3x^2)^2 \)[/tex]:
Begin by expanding the square:
- [tex]\((4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2\)[/tex].
- Calculating each term gives [tex]\(16x^8 - 24x^6 + 9x^4\)[/tex].
2. Multiply the simplified expression by [tex]\(3x\)[/tex]:
Distribute [tex]\(3x\)[/tex] across each term:
- [tex]\(3x \cdot 16x^8 = 48x^9\)[/tex],
- [tex]\(3x \cdot (-24x^6) = -72x^7\)[/tex],
- [tex]\(3x \cdot 9x^4 = 27x^5\)[/tex].
3. Combine these with the original [tex]\(9x^5\)[/tex]:
Add the polynomial from the previous step to [tex]\(9x^5\)[/tex]:
- The terms are [tex]\(48x^9 - 72x^7 + 27x^5 + 9x^5\)[/tex].
- Combine like terms: [tex]\(27x^5 + 9x^5 = 36x^5\)[/tex].
So, the expression [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex] simplifies to [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
Thus, the correct equivalent expression is:
[tex]\[48x^9 - 72x^7 + 36x^5\][/tex]
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