For the polynomial [tex]\(-2m^2n^3 + 2m^n n^3 + 7n^2 - 6m^4\)[/tex] to be a binomial with a degree of 4 after it has been fully simplified, which must be the missing exponent on the [tex]\(m\)[/tex] in the second term?

A. 0
B. 1
C. 2
D. 4



Answer :

Alright, let's break down the given polynomial step-by-step to determine the missing exponent on the [tex]\( m \)[/tex] in the term [tex]\( 2 m^2 n^3 \)[/tex].

The given polynomial is:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]

First, simplify the polynomial by combining like terms:

1. Combine the terms [tex]\(-2 m^2 n^3\)[/tex] and [tex]\( 2 m^2 n^3\)[/tex]:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 = 0 \][/tex]

2. After combining these terms, we are left with:
[tex]\[ 0 + 7 n^2 - 6 m^4 \][/tex]
[tex]\[ = 7 n^2 - 6 m^4 \][/tex]

Now we have a simplified polynomial:
[tex]\[ 7 n^2 - 6 m^4 \][/tex]

We can see that this simplified polynomial is a binomial (a polynomial with two terms) consisting of [tex]\( 7 n^2 \)[/tex] and [tex]\( -6 m^4 \)[/tex].

Next, let's examine the degrees of each term in the simplified polynomial:

- The degree of [tex]\( 7 n^2 \)[/tex] is 2 (since [tex]\( n^2 \)[/tex] has an exponent of 2).
- The degree of [tex]\( -6 m^4 \)[/tex] is 4 (since [tex]\( m^4 \)[/tex] has an exponent of 4).

To ensure that the entire polynomial has the highest degree of 4 (the degree of [tex]\( -6 m^4 \)[/tex]), we need to determine which exponent on the [tex]\( m \)[/tex] in the term [tex]\( 2 m^2 n^3 \)[/tex] must be adjusted.

By examining the original polynomial before simplification:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]

Given that the simplified polynomial must result in a degree of 4 and identifying the highest degree in the term [tex]\( -6 m^4 \)[/tex], we infer that the missing exponent on the [tex]\( m \)[/tex] term (in [tex]\( 2 m^2 n^3 \)[/tex]) required to create such a default highest degree is:

[tex]\[ \boxed{4} \][/tex]