Question

which terms could be used as the first term of the expression below to create a polynomial with a degree of 5 written in standard form? check all that apply. \\( -4x^{3}y^{2} \\) \\( x^{3} \\) \\( 8.4x^{4}y^{2} \\) \\( 5x^{4}y \\) \\( -xy^{3} \\) \\( \frac{-2x^{4}}{y} \\) \\( +2x^{2}y^{2} - 3xy^{3} \\)

Explanation:

Step1: Define polynomial degree rule

The degree of a term with variables $x$ and $y$ is the sum of the exponents of $x$ and $y$. For the whole polynomial to have degree 5, the first term must have degree 5, and be the highest degree term (standard form orders terms from highest to lowest degree).

Step2: Calculate degree of each option

• For $-4x^3y^2$: $3+2=5$
• For $x^3$: $3+0=3$
• For $8.4x^4y^2$: $4+2=6$
• For $5x^4y$: $4+1=5$
• For $-xy^3$: $1+3=4$
• For $\frac{-2x^4}{y}$: This is not a polynomial term (negative exponent on $y$)

Step3: Validate polynomial and degree

We need a degree 5 term that is a valid polynomial term, and ensures the polynomial is in standard form (so the term's degree is the highest, which 5 is higher than the existing terms' degrees: $2x^2y^2$ has degree 4, $-3xy^3$ has degree 4).

Answer:

• $-4x^3y^2$
• $5x^4y$