Subtract the polynomials and indicate the degree of the result: (10x^4y^2 - 11x^3y - 3y) - (9x^4y^2 + 4x^3y + 11y - 8x).

Step-by-step solution

1. 1

   Distribute the minus sign across all four terms of the second polynomial. Every sign inside the second bracket flips, including the $-8x$, which becomes $+8x$. That last one is where most errors happen because it is already negative:

   $10x^{4}y^{2}-11x^{3}y-3y-9x^{4}y^{2}-4x^{3}y-11y+8x$

2. 2

   Identify like terms in two variables. Terms are alike only when both exponents match, so $x^{4}y^{2}$ pairs only with $x^{4}y^{2}$, and $x^{3}y$ only with $x^{3}y$. The $y$ terms pair with each other, and $8x$ has no partner at all.

3. 3

   Combine the matching terms.

   $10x^{4}y^{2}-9x^{4}y^{2}=x^{4}y^{2}$
   $-11x^{3}y-4x^{3}y=-15x^{3}y$
   $-3y-11y=-14y$

   and the lone term $8x$ carries through unchanged, giving

   $x^{4}y^{2}-15x^{3}y-14y+8x$

4. 4

   Compute the degree by summing exponents within each term. For a multivariable polynomial the degree of a term is the sum of its exponents, and the degree of the polynomial is the largest such sum:

   $x^{4}y^{2}:\ 4+2=6,\qquad -15x^{3}y:\ 3+1=4,\qquad -14y:\ 1,\qquad 8x:\ 1$

   The maximum is $6$.

5. 5

   State the answer and verify numerically. The difference is $x^{4}y^{2}-15x^{3}y-14y+8x$ with degree $6$. Substituting $(x,y)=(1.3,0.7)$ into the original expression gives $-21.069011$ and the simplified form gives the same $-21.069011$; at $(2,-1.5)$ both give exactly $253$. The two independent test points confirm every coefficient.