High School

Which expression is equivalent to [tex]$9x^5 + 3x(4x^4 - 3x^2)^2$[/tex]?

A. [tex]$48x^9 - 24x^6 + 9x^5 + 9x^4$[/tex]
B. [tex][tex]$48x^9 + 9x^5 - 9x^4$[/tex][/tex]
C. [tex]$48x^9 + 36x^5$[/tex]
D. [tex]$48x^9 - 72x^7 + 36x^5$[/tex]

Answer :

Sure, let's determine which given expression is equivalent to [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex].

First, let's rewrite and simplify the given expression step-by-step:

1. Start with the given expression:
[tex]\[
9x^5 + 3x(4x^4 - 3x^2)^2
\][/tex]

2. Expand the squared term [tex]\((4x^4 - 3x^2)^2\)[/tex]:
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4 - 3x^2)(4x^4 - 3x^2)
\][/tex]

3. Multiply out the terms inside the parentheses:
[tex]\[
(4x^4 - 3x^2)(4x^4 - 3x^2) = 4x^4 \cdot 4x^4 + 4x^4 \cdot (-3x^2) + (-3x^2) \cdot 4x^4 + (-3x^2) \cdot (-3x^2)
\][/tex]
[tex]\[
= 16x^8 - 12x^6 - 12x^6 + 9x^4
\][/tex]
[tex]\[
= 16x^8 - 24x^6 + 9x^4
\][/tex]

4. Substitute the expanded form back into the original expression:
[tex]\[
9x^5 + 3x(16x^8 - 24x^6 + 9x^4)
\][/tex]

5. Distribute [tex]\(3x\)[/tex] over the terms inside the parentheses:
[tex]\[
9x^5 + 3x \cdot 16x^8 + 3x \cdot (-24x^6) + 3x \cdot 9x^4
\][/tex]
[tex]\[
= 9x^5 + 48x^9 - 72x^7 + 27x^5
\][/tex]

6. Combine like terms:
[tex]\[
9x^5 + 27x^5 + 48x^9 - 72x^7
\][/tex]
[tex]\[
= 36x^5 + 48x^9 - 72x^7
\][/tex]

7. Match to the given choices, you are left with:
[tex]\[
\boxed{48x^9 - 72x^7 + 36x^5}
\][/tex]

Therefore, the expression [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex] is equivalent to [tex]\( 48x^9 - 72x^7 + 36x^5 \)[/tex].

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