marcus states that the polynomial expression $3x^{3}-4x^{2}y + y^{3}+2$ is in standard form. ariel states…

marcus states that the polynomial expression $3x^{3}-4x^{2}y + y^{3}+2$ is in standard form. ariel states that it should be $y^{3}-4x^{2}y + 3x^{3}+2$. explain which student is correct and why.

marcus states that the polynomial expression $3x^{3}-4x^{2}y + y^{3}+2$ is in standard form. ariel states that it should be $y^{3}-4x^{2}y + 3x^{3}+2$. explain which student is correct and why.

Answer

Explanation:

Step1: Recall standard - form rule

A polynomial in two variables (x) and (y) is in standard form when the terms are arranged in descending order of the sum of the exponents of the variables in each term.

Step2: Calculate exponents sums

For the term (3x^{3}), the sum of exponents is (3). For (-4x^{2}y), the sum of exponents is (2 + 1=3). For (y^{3}), the sum of exponents is (3). When the sum of exponents is the same, we usually arrange them in alphabetical order of the variables ((x) before (y)).

Step3: Determine standard form

The correct standard - form is (3x^{3}-4x^{2}y + y^{3}+2) since (x) comes before (y) alphabetically when the degree of the terms is the same.

Answer:

Marcus is correct because a polynomial in two variables (x) and (y) with terms of the same degree (sum of exponents) is arranged in alphabetical order of the variables ((x) before (y)), and (3x^{3}-4x^{2}y + y^{3}+2) follows this rule for standard form.