Answer :

To determine the degree of the polynomial

[tex]$$
4x^6 - 3w^2x^2u - u^5w^2 + 7,
$$[/tex]

we must find the degree of each individual term and then take the maximum.

1. For the term [tex]$$4x^6,$$[/tex] the only variable is [tex]$x$[/tex] with an exponent of 6. Therefore, its degree is

[tex]$$
6.
$$[/tex]

2. For the term [tex]$$-3w^2x^2u,$$[/tex] there are three variables: [tex]$w$[/tex], [tex]$x$[/tex], and [tex]$u$[/tex] with exponents 2, 2, and 1 respectively. The total degree is

[tex]$$
2 + 2 + 1 = 5.
$$[/tex]

3. For the term [tex]$$-u^5w^2,$$[/tex] there are two variables: [tex]$u$[/tex] raised to the power of 5 and [tex]$w$[/tex] raised to the power of 2. The total degree is

[tex]$$
5 + 2 = 7.
$$[/tex]

4. The last term, [tex]$$7,$$[/tex] is a constant and has degree

[tex]$$
0.
$$[/tex]

Now we compare the degrees of all terms:

[tex]$$
\max(6, 5, 7, 0) = 7.
$$[/tex]

Thus, the degree of the polynomial is

[tex]$$
7.
$$[/tex]

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