Find the degree of the polynomial:
$4 a^{4} - 2 (3 + a^{4} - a b + 4 b^{3}) - 2 a^{4}$

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Solution

Simplifying the expression,
$4 a^{4} - 2 (3 + a^{4} - a b + 4 b^{3}) - 2 a^{4}$
$= 4 a^{4} - 6 - 2 a^{4} + 2 a b - 8 b^{3} - 2 a^{4}$
Grouping the like terms we get,
$\Rightarrow (4 a^{4} - 2 a^{4} - 2 a^{4}) - 6 + 2 a b - 8 b^{3}$
$= (4 - 2 - 2) a^{4} - 6 + 2 a b - 8 b^{3}$
$= 0 a^{4} - 6 + 2 a b - 8 b^{3}$
$= - 6 + 2 a b - 8 b^{3}$
Hence, the given expression reduces to $- 6 + 2 a b - 8 b^{3}$
Identifying the degree of each monomial,

$- 6 = - 6 a^{0} \to Degree = 0$

$2 a b = 2 a^{1} b^{1} \to Degree = 1 + 1 = 2$

$- 8 b^{3} \to Degree = 3$

$∴$ Among these monomials, maximum degree is $3$ . Hence, the degree of the polynomial is $3$ .

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