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Answer :
Let's solve the problem step-by-step to find which expression is equivalent to [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex].
1. Identify the Expression:
We have the expression:
[tex]\[
9x^5 + 3x(4x^4 - 3x^2)^2
\][/tex]
2. Expand the Expression Inside the Parentheses:
First, expand [tex]\((4x^4 - 3x^2)^2\)[/tex]:
[tex]\[
(4x^4 - 3x^2)(4x^4 - 3x^2)
\][/tex]
Using the distributive property (FOIL method), this becomes:
[tex]\[
(4x^4)(4x^4) - (4x^4)(3x^2) - (3x^2)(4x^4) + (3x^2)(3x^2)
\][/tex]
[tex]\[
= 16x^8 - 12x^6 - 12x^6 + 9x^4
\][/tex]
[tex]\[
= 16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply by 3x:
Now multiply the expanded expression by [tex]\(3x\)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4)
\][/tex]
Distribute the [tex]\(3x\)[/tex] across each term:
[tex]\[
= 48x^9 - 72x^7 + 27x^5
\][/tex]
4. Combine with the Original Term:
Add this result to the original first term of the expression:
[tex]\[
9x^5 + 48x^9 - 72x^7 + 27x^5
\][/tex]
Combine like terms:
[tex]\[
= 48x^9 - 72x^7 + (9x^5 + 27x^5)
\][/tex]
[tex]\[
= 48x^9 - 72x^7 + 36x^5
\][/tex]
Now, we compare the simplified expression [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex] with the given options:
- [tex]\(48x^9 - 24x^6 + 9x^5 + 9x^4\)[/tex]
- [tex]\(48x^9 + 9x^5 - 9x^4\)[/tex]
- [tex]\(48x^9 + 36x^5\)[/tex]
- [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
The equivalent expression is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{48x^9 - 72x^7 + 36x^5}
\][/tex]
1. Identify the Expression:
We have the expression:
[tex]\[
9x^5 + 3x(4x^4 - 3x^2)^2
\][/tex]
2. Expand the Expression Inside the Parentheses:
First, expand [tex]\((4x^4 - 3x^2)^2\)[/tex]:
[tex]\[
(4x^4 - 3x^2)(4x^4 - 3x^2)
\][/tex]
Using the distributive property (FOIL method), this becomes:
[tex]\[
(4x^4)(4x^4) - (4x^4)(3x^2) - (3x^2)(4x^4) + (3x^2)(3x^2)
\][/tex]
[tex]\[
= 16x^8 - 12x^6 - 12x^6 + 9x^4
\][/tex]
[tex]\[
= 16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply by 3x:
Now multiply the expanded expression by [tex]\(3x\)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4)
\][/tex]
Distribute the [tex]\(3x\)[/tex] across each term:
[tex]\[
= 48x^9 - 72x^7 + 27x^5
\][/tex]
4. Combine with the Original Term:
Add this result to the original first term of the expression:
[tex]\[
9x^5 + 48x^9 - 72x^7 + 27x^5
\][/tex]
Combine like terms:
[tex]\[
= 48x^9 - 72x^7 + (9x^5 + 27x^5)
\][/tex]
[tex]\[
= 48x^9 - 72x^7 + 36x^5
\][/tex]
Now, we compare the simplified expression [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex] with the given options:
- [tex]\(48x^9 - 24x^6 + 9x^5 + 9x^4\)[/tex]
- [tex]\(48x^9 + 9x^5 - 9x^4\)[/tex]
- [tex]\(48x^9 + 36x^5\)[/tex]
- [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
The equivalent expression is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{48x^9 - 72x^7 + 36x^5}
\][/tex]
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