Question

julian fully simplifies this polynomial and then writes it in standard form.
$4x^{2}y^{2} - 2y^{4} - 8xy^{3} + 9x^{3}y + 6y^{4} - 2xy^{3} - 3x^{4} + x^{2}y^{2}$
if julian wrote the last term as $-3x^{4}$, which must be the first term of his polynomial in standard form?
$\circ\\ 4y^{4}$
$\circ\\ 6y^{4}$
$\circ\\ -2xy^{3}$
$\circ\\ -10xy^{3}$

Explanation:

Step1: Combine like terms for \(y^4\)

The terms with \(y^4\) are \(-2y^4\) and \(6y^4\). Combining them: \(-2y^4 + 6y^4 = 4y^4\)

Step2: Combine like terms for \(xy^3\)

The terms with \(xy^3\) are \(-8xy^3\) and \(-2xy^3\). Combining them: \(-8xy^3 - 2xy^3 = -10xy^3\)

Step3: Combine like terms for \(x^2y^2\)

The terms with \(x^2y^2\) are \(4x^2y^2\) and \(x^2y^2\). Combining them: \(4x^2y^2 + x^2y^2 = 5x^2y^2\)

Step4: Identify the term with the highest degree

The polynomial in standard form (when last term is \(-3x^4\)) should be ordered by descending degree. The term \(4y^4\) has degree 4 (since \(y^4\) has degree 4), \(9x^3y\) has degree 4 (3 + 1 = 4), \(-10xy^3\) has degree 4 (1 + 3 = 4), \(5x^2y^2\) has degree 4 (2 + 2 = 4), and \(-3x^4\) has degree 4. But when ordering, we look at the leading term. Wait, no—wait, the last term is \(-3x^4\), so the polynomial is ordered such that the first term is the one with the highest degree and appropriate ordering. Wait, actually, when combining like terms, the term \(4y^4\) is a degree 4 term. Let's re - check the combination of \(y^4\) terms: \(-2y^4+6y^4 = 4y^4\). Now, when writing in standard form, if the last term is \(-3x^4\), we order the terms by descending degree. The term \(4y^4\) is a degree 4 term. Let's check the options. The first term should be the term with the highest degree and when we combined the \(y^4\) terms we got \(4y^4\).

Answer:

A. \(4y^4\)