Answer :
To solve the problem of finding which expression is equivalent to [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex], we'll expand and simplify the expression step by step to match one of the options given.
1. Write down the original expression:
The expression is [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex].
2. Focus on the term inside the parentheses and expand it:
The expression inside the parentheses is [tex]\((4x^4 - 3x^2)^2\)[/tex].
a. First, apply the binomial square formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
b. Here, [tex]\(a = 4x^4\)[/tex] and [tex]\(b = 3x^2\)[/tex].
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2
\][/tex]
c. Calculate each term:
- [tex]\((4x^4)^2 = 16x^8\)[/tex]
- [tex]\(-2(4x^4)(3x^2) = -24x^6\)[/tex]
- [tex]\((3x^2)^2 = 9x^4\)[/tex]
d. Combine these to get:
[tex]\[
(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply the expanded expression by [tex]\(3x\)[/tex]:
We now multiply each term by [tex]\(3x\)[/tex]:
a. [tex]\(3x \cdot 16x^8 = 48x^9\)[/tex]
b. [tex]\(3x \cdot (-24x^6) = -72x^7\)[/tex]
c. [tex]\(3x \cdot 9x^4 = 27x^5\)[/tex]
Summing these, the result of multiplying [tex]\(3x\)[/tex] by the expanded expression is:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]
4. Add this to the remaining term of the original expression:
Add the result above to [tex]\(9x^5\)[/tex]:
[tex]\[
48x^9 - 72x^7 + 27x^5 + 9x^5
\][/tex]
5. Combine like terms:
Combine [tex]\(27x^5 + 9x^5\)[/tex] to get [tex]\(36x^5\)[/tex].
The expression simplifies to:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
6. Compare with given options:
Finally, we compare the expanded expression with the given options:
The correct equivalent expression is:
[tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
This matches the fourth option provided in the problem.
1. Write down the original expression:
The expression is [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex].
2. Focus on the term inside the parentheses and expand it:
The expression inside the parentheses is [tex]\((4x^4 - 3x^2)^2\)[/tex].
a. First, apply the binomial square formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
b. Here, [tex]\(a = 4x^4\)[/tex] and [tex]\(b = 3x^2\)[/tex].
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2
\][/tex]
c. Calculate each term:
- [tex]\((4x^4)^2 = 16x^8\)[/tex]
- [tex]\(-2(4x^4)(3x^2) = -24x^6\)[/tex]
- [tex]\((3x^2)^2 = 9x^4\)[/tex]
d. Combine these to get:
[tex]\[
(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply the expanded expression by [tex]\(3x\)[/tex]:
We now multiply each term by [tex]\(3x\)[/tex]:
a. [tex]\(3x \cdot 16x^8 = 48x^9\)[/tex]
b. [tex]\(3x \cdot (-24x^6) = -72x^7\)[/tex]
c. [tex]\(3x \cdot 9x^4 = 27x^5\)[/tex]
Summing these, the result of multiplying [tex]\(3x\)[/tex] by the expanded expression is:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]
4. Add this to the remaining term of the original expression:
Add the result above to [tex]\(9x^5\)[/tex]:
[tex]\[
48x^9 - 72x^7 + 27x^5 + 9x^5
\][/tex]
5. Combine like terms:
Combine [tex]\(27x^5 + 9x^5\)[/tex] to get [tex]\(36x^5\)[/tex].
The expression simplifies to:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
6. Compare with given options:
Finally, we compare the expanded expression with the given options:
The correct equivalent expression is:
[tex]\(48x^9 - 72x^7 + 36x^5\)[/tex]
This matches the fourth option provided in the problem.
