Question

which represents the polynomial written in standard form? $8x^{2}y^{2}-3x^{3}y + 4x^{4}-7xy^{3}$\
options:\
$4x^{4}-3x^{3}y + 8x^{2}y^{2}-7xy^{3}$\
$-7xy^{3}-3x^{3}y + 8x^{2}y^{2}+4x^{4}$\
$4x^{4}-7xy^{3}-3x^{3}y + 8x^{2}y^{2}$

Explanation:

Step1: Define standard polynomial form

A polynomial in two variables is in standard form when terms are ordered by descending total degree (sum of exponents of $x$ and $y$), and for equal degrees, by descending exponent of $x$.

Step2: Calculate term degrees

• For $8x^2y^2$: $2+2=4$
• For $-3x^3y$: $3+1=4$
• For $4x^4$: $4+0=4$
• For $-7xy^3$: $1+3=4$

Step3: Order by descending $x$-exponent

For terms with total degree 4, sort by $x$-exponent from highest to lowest:

1. $4x^4$ ($x$-exponent = 4)
2. $-3x^3y$ ($x$-exponent = 3)
3. $8x^2y^2$ ($x$-exponent = 2)
4. $-7xy^3$ ($x$-exponent = 1)

Answer:

$4x^4 - 3x^3y + 8x^2y^2 - 7xy^3$