Answer :
To determine which expression is equivalent to [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex], let's go through the steps to simplify it.
### Step 1: Expand the Inner Expression
First, let's examine the expression inside the parentheses: [tex]\( (4x^4 - 3x^2)^2 \)[/tex].
Using the binomial expansion formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we can expand:
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2 \cdot 4x^4 \cdot 3x^2 + (3x^2)^2
\][/tex]
Calculating 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]
So, the expansion is:
[tex]\[
16x^8 - 24x^6 + 9x^4
\][/tex]
### Step 2: Multiply the Entire Expression by [tex]\(3x\)[/tex]
Now that we have expanded the inner expression, we multiply it by [tex]\(3x\)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4) = 3x \times 16x^8 - 3x \times 24x^6 + 3x \times 9x^4
\][/tex]
Calculating each term:
- [tex]\(3x \times 16x^8 = 48x^9\)[/tex]
- [tex]\(3x \times 24x^6 = 72x^7\)[/tex]
- [tex]\(3x \times 9x^4 = 27x^5\)[/tex]
So, this gives us:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]
### Step 3: Combine with the Original Term
Now, we need to combine this result with the original term [tex]\(9x^5\)[/tex]:
[tex]\[
9x^5 + (48x^9 - 72x^7 + 27x^5) = 48x^9 - 72x^7 + (27x^5 + 9x^5)
\][/tex]
Combine the like terms:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
### Conclusion
The expression [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex] matches one of the given options. Therefore, the correct answer is:
[tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
### Step 1: Expand the Inner Expression
First, let's examine the expression inside the parentheses: [tex]\( (4x^4 - 3x^2)^2 \)[/tex].
Using the binomial expansion formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we can expand:
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2 \cdot 4x^4 \cdot 3x^2 + (3x^2)^2
\][/tex]
Calculating 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]
So, the expansion is:
[tex]\[
16x^8 - 24x^6 + 9x^4
\][/tex]
### Step 2: Multiply the Entire Expression by [tex]\(3x\)[/tex]
Now that we have expanded the inner expression, we multiply it by [tex]\(3x\)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4) = 3x \times 16x^8 - 3x \times 24x^6 + 3x \times 9x^4
\][/tex]
Calculating each term:
- [tex]\(3x \times 16x^8 = 48x^9\)[/tex]
- [tex]\(3x \times 24x^6 = 72x^7\)[/tex]
- [tex]\(3x \times 9x^4 = 27x^5\)[/tex]
So, this gives us:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]
### Step 3: Combine with the Original Term
Now, we need to combine this result with the original term [tex]\(9x^5\)[/tex]:
[tex]\[
9x^5 + (48x^9 - 72x^7 + 27x^5) = 48x^9 - 72x^7 + (27x^5 + 9x^5)
\][/tex]
Combine the like terms:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
### Conclusion
The expression [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex] matches one of the given options. Therefore, the correct answer is:
[tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
