What is true about the sum of the two polynomials 6s^2t - 2st^2 and 4s^2t - 3st^2? Is it a binomial or a trinomial, and what is its degree?

Step-by-step solution

1. 1

   Identify which terms are alike. Two terms are like terms only if every variable carries the same exponent. Here $6s^2t$ and $4s^2t$ match ($s^2t$), and $-2st^2$ and $-3st^2$ match ($st^2$). But $s^2t$ and $st^2$ are not alike, so they can never be merged.

2. 2

   Add the coefficients of each pair.

   $6s^2t+4s^2t=10s^2t,\qquad -2st^2+(-3st^2)=-5st^2$

3. 3

   Write the sum and count its terms.

   $10s^2t-5st^2$

   Two unlike terms survive, so the sum is a binomial — not a trinomial, and it does not collapse to a monomial.

4. 4

   Find the degree correctly. In a multivariable term the degree is the sum of the exponents:

   $\deg(10s^2t)=2+1=3,\qquad \deg(-5st^2)=1+2=3$

   The polynomial's degree is the largest of these, so the degree is $3$. Reading the degree as $2$ from the visible exponent on $s$ is the standard trap in this question.

5. 5

   Check with a substitution. At $s=2,\ t=1$: the two originals give $6(4)(1)-2(2)(1)=20$ and $4(4)(1)-3(2)(1)=10$, summing to $30$. The result $10s^2t-5st^2$ gives $10(4)(1)-5(2)(1)=40-10=30\ \checkmark$.