Answer :
To find the expression that is equivalent to [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex], follow these steps:
1. Identify the Given Expression:
The expression we need to simplify and expand is:
[tex]\[
9x^5 + 3x(4x^4 - 3x^2)^2
\][/tex]
2. Expand the Inner Expression:
Focus first on expanding [tex]\((4x^4 - 3x^2)^2\)[/tex].
Using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we substitute [tex]\(a = 4x^4\)[/tex] and [tex]\(b = 3x^2\)[/tex]:
- [tex]\(a^2 = (4x^4)^2 = 16x^8\)[/tex]
- [tex]\(-2ab = -2 \times 4x^4 \times 3x^2 = -24x^6\)[/tex]
- [tex]\(b^2 = (3x^2)^2 = 9x^4\)[/tex]
Put these together:
[tex]\[
(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply the Expression by [tex]\(3x\)[/tex]:
Now substitute back into the original expression, and multiply the expanded form by [tex]\(3x\)[/tex].
[tex]\[
3x(16x^8 - 24x^6 + 9x^4) = 48x^9 - 72x^7 + 27x^5
\][/tex]
4. Combine the Terms:
Add the other part of the initial expression [tex]\(9x^5\)[/tex].
[tex]\[
9x^5 + 48x^9 - 72x^7 + 27x^5
\][/tex]
Combine like terms:
- [tex]\(9x^5 + 27x^5 = 36x^5\)[/tex]
Therefore, the expression simplifies to:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
5. Select the Correct Option:
Comparing with the given options, the expression [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex] matches the last option:
- [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
Thus, the equivalent expression is [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
1. Identify the Given Expression:
The expression we need to simplify and expand is:
[tex]\[
9x^5 + 3x(4x^4 - 3x^2)^2
\][/tex]
2. Expand the Inner Expression:
Focus first on expanding [tex]\((4x^4 - 3x^2)^2\)[/tex].
Using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex], we substitute [tex]\(a = 4x^4\)[/tex] and [tex]\(b = 3x^2\)[/tex]:
- [tex]\(a^2 = (4x^4)^2 = 16x^8\)[/tex]
- [tex]\(-2ab = -2 \times 4x^4 \times 3x^2 = -24x^6\)[/tex]
- [tex]\(b^2 = (3x^2)^2 = 9x^4\)[/tex]
Put these together:
[tex]\[
(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply the Expression by [tex]\(3x\)[/tex]:
Now substitute back into the original expression, and multiply the expanded form by [tex]\(3x\)[/tex].
[tex]\[
3x(16x^8 - 24x^6 + 9x^4) = 48x^9 - 72x^7 + 27x^5
\][/tex]
4. Combine the Terms:
Add the other part of the initial expression [tex]\(9x^5\)[/tex].
[tex]\[
9x^5 + 48x^9 - 72x^7 + 27x^5
\][/tex]
Combine like terms:
- [tex]\(9x^5 + 27x^5 = 36x^5\)[/tex]
Therefore, the expression simplifies to:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
5. Select the Correct Option:
Comparing with the given options, the expression [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex] matches the last option:
- [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
Thus, the equivalent expression is [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].
