Subtract the polynomials (x^3 - 7xy - 9y^2) - (2x^3 + 3xy - 7y^2) and state the degree of the resulting polynomial.

Step-by-step solution

1. 1

   Change the subtraction into an addition. Subtracting a polynomial means adding its opposite, so every sign inside the second bracket flips:

   $-\left(2x^{3}+3xy-7y^{2}\right)=-2x^{3}-3xy+7y^{2}.$

   The $-7y^2$ becoming $+7y^2$ is the term most often left unchanged by mistake.

2. 2

   Write the expression with no brackets.

   $x^{3}-7xy-9y^{2}-2x^{3}-3xy+7y^{2}.$

3. 3

   Group like terms. In two variables, like terms must match in both exponents:

   $\left(x^{3}-2x^{3}\right)+\left(-7xy-3xy\right)+\left(-9y^{2}+7y^{2}\right).$

4. 4

   Combine each group.

   $-x^{3}-10xy-2y^{2}.$

5. 5

   Read off the degree. The degree of a multivariable term is the sum of its exponents: $x^3$ has degree $3$, $xy$ has degree $1+1=2$, and $y^2$ has degree $2$. The largest is $3$, and its term did not cancel, so the difference is a degree 3 polynomial.

6. 6

   Spot-check with numbers. At $x=1,y=1$ the original is $(1-7-9)-(2+3-7)=-15-(-2)=-13$, and the answer gives $-1-10-2=-13$. At $x=2,y=-1$ the original is $(8+14-9)-(16-6-7)=13-3=10$, and the answer gives $-8+20-2=10$.