Question

for the polynomial $-2m^{2}n^{3} + 2m^{?}n^{3} + 7n^{2} - 6m^{4}$ to be a binomial with a degree of 4 after it has been fully simplified, which must be the missing exponent on the $m$ in the second term?
○ 0
○ 1
○ 2
○ 4

Explanation:

Step1: Define missing exponent

Let the missing exponent be $x$, so the polynomial is $-2m^2n^3 + 2m^xn^3 + 7n^2 - 6m^4$.

Step2: Combine like terms for binomial

To get a binomial, two pairs of terms must cancel/combine. The only like terms are $-2m^2n^3$ and $2m^xn^3$. For them to combine, $x=2$, making the sum: $-2m^2n^3 + 2m^2n^3 = 0$.

Step3: Check degree of result

After combining, the polynomial becomes $7n^2 - 6m^4$. The degree of $-6m^4$ is 4, which matches the required degree.

Answer:

2 (Option C)