Answer :
To simplify the given expression [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex] and determine which of the provided options it is equivalent to, we can break down the problem into smaller parts:
1. Simplify the Inner Expression:
The expression inside the parentheses is [tex]\( 4x^4 - 3x^2 \)[/tex].
2. Square the Inner Expression:
We want to find [tex]\((4x^4 - 3x^2)^2\)[/tex].
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4 - 3x^2)(4x^4 - 3x^2)
\][/tex]
Use the distributive property (FOIL method):
[tex]\[
(4x^4)(4x^4) + (4x^4)(-3x^2) + (-3x^2)(4x^4) + (-3x^2)(-3x^2)
\][/tex]
Simplify each term:
- [tex]\( (4x^4)(4x^4) = 16x^8 \)[/tex]
- [tex]\( (4x^4)(-3x^2) = -12x^6 \)[/tex]
- [tex]\( (-3x^2)(4x^4) = -12x^6 \)[/tex]
- [tex]\( (-3x^2)(-3x^2) = 9x^4 \)[/tex]
Combine like terms:
[tex]\[
16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply by [tex]\( 3x \)[/tex]:
Next, multiply the squared expression [tex]\( 16x^8 - 24x^6 + 9x^4 \)[/tex] by [tex]\( 3x \)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4)
\][/tex]
Distribute [tex]\( 3x \)[/tex]:
- [tex]\( 3x \cdot 16x^8 = 48x^9 \)[/tex]
- [tex]\( 3x \cdot (-24x^6) = -72x^7 \)[/tex]
- [tex]\( 3x \cdot 9x^4 = 27x^5 \)[/tex]
This gives us:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]
4. Add [tex]\( 9x^5 \)[/tex] to This Result:
Add [tex]\( 9x^5 \)[/tex] from the original expression:
[tex]\[
48x^9 - 72x^7 + 27x^5 + 9x^5 = 48x^9 - 72x^7 + 36x^5
\][/tex]
The expanded expression is [tex]\( 48x^9 - 72x^7 + 36x^5 \)[/tex].
Therefore, the expression is equivalent to the option:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
1. Simplify the Inner Expression:
The expression inside the parentheses is [tex]\( 4x^4 - 3x^2 \)[/tex].
2. Square the Inner Expression:
We want to find [tex]\((4x^4 - 3x^2)^2\)[/tex].
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4 - 3x^2)(4x^4 - 3x^2)
\][/tex]
Use the distributive property (FOIL method):
[tex]\[
(4x^4)(4x^4) + (4x^4)(-3x^2) + (-3x^2)(4x^4) + (-3x^2)(-3x^2)
\][/tex]
Simplify each term:
- [tex]\( (4x^4)(4x^4) = 16x^8 \)[/tex]
- [tex]\( (4x^4)(-3x^2) = -12x^6 \)[/tex]
- [tex]\( (-3x^2)(4x^4) = -12x^6 \)[/tex]
- [tex]\( (-3x^2)(-3x^2) = 9x^4 \)[/tex]
Combine like terms:
[tex]\[
16x^8 - 24x^6 + 9x^4
\][/tex]
3. Multiply by [tex]\( 3x \)[/tex]:
Next, multiply the squared expression [tex]\( 16x^8 - 24x^6 + 9x^4 \)[/tex] by [tex]\( 3x \)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4)
\][/tex]
Distribute [tex]\( 3x \)[/tex]:
- [tex]\( 3x \cdot 16x^8 = 48x^9 \)[/tex]
- [tex]\( 3x \cdot (-24x^6) = -72x^7 \)[/tex]
- [tex]\( 3x \cdot 9x^4 = 27x^5 \)[/tex]
This gives us:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]
4. Add [tex]\( 9x^5 \)[/tex] to This Result:
Add [tex]\( 9x^5 \)[/tex] from the original expression:
[tex]\[
48x^9 - 72x^7 + 27x^5 + 9x^5 = 48x^9 - 72x^7 + 36x^5
\][/tex]
The expanded expression is [tex]\( 48x^9 - 72x^7 + 36x^5 \)[/tex].
Therefore, the expression is equivalent to the option:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
