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Answer :
Sure! Let's find an expression equivalent to [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex].
We will simplify the expression step-by-step.
1. Start with the expression [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex].
2. First, simplify the term inside the parentheses:
[tex]\[
(4x^4 - 3x^2)^2
\][/tex]
3. To expand [tex]\( (4x^4 - 3x^2)^2 \)[/tex], use the binomial theorem:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
where [tex]\( a = 4x^4 \)[/tex] and [tex]\( b = 3x^2 \)[/tex].
Thus,
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2
\][/tex]
4. Simplify 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]
5. Substitute back into the expanded form:
[tex]\[
(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4
\][/tex]
6. Multiply this result by [tex]\( 3x \)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4) = 48x^9 - 72x^7 + 27x^5
\][/tex]
7. Add this to the original expression:
[tex]\[
9x^5 + (48x^9 - 72x^7 + 27x^5)
\][/tex]
8. Combine like terms:
[tex]\[
9x^5 + 27x^5 + 48x^9 - 72x^7 = 48x^9 - 72x^7 + 36x^5
\][/tex]
So, the expression equivalent to [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex] is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
Therefore, the correct answer is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
which corresponds to option 4.
We will simplify the expression step-by-step.
1. Start with the expression [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex].
2. First, simplify the term inside the parentheses:
[tex]\[
(4x^4 - 3x^2)^2
\][/tex]
3. To expand [tex]\( (4x^4 - 3x^2)^2 \)[/tex], use the binomial theorem:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
where [tex]\( a = 4x^4 \)[/tex] and [tex]\( b = 3x^2 \)[/tex].
Thus,
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2
\][/tex]
4. Simplify 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]
5. Substitute back into the expanded form:
[tex]\[
(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4
\][/tex]
6. Multiply this result by [tex]\( 3x \)[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4) = 48x^9 - 72x^7 + 27x^5
\][/tex]
7. Add this to the original expression:
[tex]\[
9x^5 + (48x^9 - 72x^7 + 27x^5)
\][/tex]
8. Combine like terms:
[tex]\[
9x^5 + 27x^5 + 48x^9 - 72x^7 = 48x^9 - 72x^7 + 36x^5
\][/tex]
So, the expression equivalent to [tex]\( 9x^5 + 3x(4x^4 - 3x^2)^2 \)[/tex] is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
Therefore, the correct answer is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]
which corresponds to option 4.
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