To determine which given polynomial is written in standard form, we need to arrange the terms in descending order of their degrees. The degree of a term is the sum of the exponents of the variables in that term. Let's identify and order the terms accordingly.
Given polynomial:
[tex]\[ 8x^2 y^2 - 3x^3 y + 4x^4 - 7xy^3 \][/tex]
First, determine the degree of each term:
1. \( 8x^2 y^2 \): degree = \( 2 + 2 = 4 \)
2. \( -3x^3 y \): degree = \( 3 + 1 = 4 \)
3. \( 4x^4 \): degree = \( 4 \)
4. \( -7xy^3 \): degree = \( 1 + 3 = 4 \)
Since all these terms have the same degree, the standard form should order the terms in descending powers of \( x \) and then \( y \) where applicable.
Now arrange them based on the powers of \( x \):
[tex]\[ 4x^4 - 3x^3 y + 8x^2 y^2 - 7xy^3 \][/tex]
Let's check which of the given options matches this arranged form:
1. \( 4x^4 - 3x^3 y + 8x^2 y^2 - 7xy^3 \)
2. \( 4x^4 - 7xy^3 - 3x^3 y + 8x^2 y^2 \)
3. \( 4x^4 + 8x^2 y^2 - 3x^3 y - 7xy^3 \)
4. \( -7xy^3 - 3x^3 y + 8x^2 y^2 + 4x^4 \)
It matches the first option.
Therefore, the polynomial written in standard form is:
[tex]\[ \boxed{4x^4 - 3x^3 y + 8x^2 y^2 - 7xy^3} \][/tex]
