Let's go through the process of fully simplifying the given polynomial and writing it in standard form:
The given polynomial is:
[tex]\[
4 x^2 y^2 - 2 y^4 - 8 x y^3 + 9 x^3 y + 6 y^4 - 2 x y^3 - 3 x^4 + x^2 y^2
\][/tex]
First, let's combine like terms:
1. Combine the [tex]\(y^4\)[/tex] terms:
[tex]\[
-2 y^4 + 6 y^4 = 4 y^4
\][/tex]
2. Combine the [tex]\(x^2 y^2\)[/tex] terms:
[tex]\[
4 x^2 y^2 + x^2 y^2 = 5 x^2 y^2
\][/tex]
3. Combine the [tex]\(x y^3\)[/tex] terms:
[tex]\[
-8 x y^3 - 2 x y^3 = -10 x y^3
\][/tex]
4. Identify the only [tex]\(x^3 y\)[/tex] term, which is:
[tex]\[
9 x^3 y
\][/tex]
5. Identify the only [tex]\(x^4\)[/tex] term, which is:
[tex]\[
-3 x^4
\][/tex]
Now, rewrite the polynomial with the combined like terms:
[tex]\[
-3 x^4 + 9 x^3 y + 5 x^2 y^2 - 10 x y^3 + 4 y^4
\][/tex]
Next, let's write this polynomial in standard form, which means arranging the terms in descending powers of [tex]\(x\)[/tex]:
[tex]\[
-3 x^4 + 9 x^3 y + 5 x^2 y^2 - 10 x y^3 + 4 y^4
\][/tex]
The polynomial in standard form is:
[tex]\[
-3 x^4 + 9 x^3 y + 5 x^2 y^2 - 10 x y^3 + 4 y^4
\][/tex]
Now, let's consider the question: If the last term of Julian's polynomial is [tex]\( -3 x^4 \)[/tex], we can observe that this is indeed the first term of the polynomial. Therefore, the first term in standard form is [tex]\( -3 x^4 \)[/tex].
Given the four options provided:
- [tex]\(4 y^4\)[/tex]
- [tex]\(6 y^4\)[/tex]
- [tex]\(-2 x y^3\)[/tex]
- [tex]\(-10 x y^3\)[/tex]
We can see that the correct first term in Julian's polynomial in standard form, which corresponds to the provided [tex]\( -3 x^4 \)[/tex] is:
[tex]\[
4 y^4
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{4 y^4}
\][/tex]
