Answer :
Sure! Let's find an expression that is equivalent to [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex]. Here's how we can break down the expression:
1. Simplify the expression inside the parenthesis:
- Start with the term [tex]\( (4x^4 - 3x^2)^2 \)[/tex].
- This represents a binomial expression squared.
2. Calculate the square of the binomial:
- [tex]\((4x^4 - 3x^2)^2 = (4x^4)^2 - 2 \cdot 4x^4 \cdot 3x^2 + (3x^2)^2\)[/tex]
- Simplify each term:
- [tex]\((4x^4)^2 = 16x^8\)[/tex]
- [tex]\(-2 \cdot 4x^4 \cdot 3x^2 = -24x^6\)[/tex]
- [tex]\((3x^2)^2 = 9x^4\)[/tex]
- Combine these to rewrite the squared expression:
- [tex]\((4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4\)[/tex]
3. Multiply the expanded 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]
4. Add the original term [tex]\(9x^5\)[/tex] to the expression:
- Combine it with the terms obtained from the distribution to get:
- [tex]\(48x^9 - 72x^7 + 27x^5 + 9x^5 = 48x^9 - 72x^7 + 36x^5\)[/tex]
Hence, the equivalent expression to the original one given is:
[tex]\[ 48x^9 - 72x^7 + 36x^5 \][/tex]
Upon comparing with the options provided, this matches the last option:
[tex]\[ 48x^9 - 72x^7 + 36x^5 \][/tex]
Therefore, the correct choice is:
[tex]\[ 48x^9 - 72x^7 + 36x^5 \][/tex]
1. Simplify the expression inside the parenthesis:
- Start with the term [tex]\( (4x^4 - 3x^2)^2 \)[/tex].
- This represents a binomial expression squared.
2. Calculate the square of the binomial:
- [tex]\((4x^4 - 3x^2)^2 = (4x^4)^2 - 2 \cdot 4x^4 \cdot 3x^2 + (3x^2)^2\)[/tex]
- Simplify each term:
- [tex]\((4x^4)^2 = 16x^8\)[/tex]
- [tex]\(-2 \cdot 4x^4 \cdot 3x^2 = -24x^6\)[/tex]
- [tex]\((3x^2)^2 = 9x^4\)[/tex]
- Combine these to rewrite the squared expression:
- [tex]\((4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4\)[/tex]
3. Multiply the expanded 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]
4. Add the original term [tex]\(9x^5\)[/tex] to the expression:
- Combine it with the terms obtained from the distribution to get:
- [tex]\(48x^9 - 72x^7 + 27x^5 + 9x^5 = 48x^9 - 72x^7 + 36x^5\)[/tex]
Hence, the equivalent expression to the original one given is:
[tex]\[ 48x^9 - 72x^7 + 36x^5 \][/tex]
Upon comparing with the options provided, this matches the last option:
[tex]\[ 48x^9 - 72x^7 + 36x^5 \][/tex]
Therefore, the correct choice is:
[tex]\[ 48x^9 - 72x^7 + 36x^5 \][/tex]
