To determine the number of solutions over the complex number system for the polynomial
[tex]\[ 7 x^5 - 33 x^4 - 4 x^2 + 3 x + 52 = 0, \][/tex]
we need to consider the fundamental theorem of algebra. This theorem states that every non-zero polynomial equation of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] solutions in the complex number system (counting multiplicities).
Here's a step-by-step explanation:
1. Identify the polynomial:
[tex]\[ 7 x^5 - 33 x^4 - 4 x^2 + 3 x + 52 = 0. \][/tex]
2. Determine the degree of the polynomial by finding the highest power of the variable [tex]\( x \)[/tex]. The polynomial is written as:
[tex]\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0, \][/tex]
where [tex]\( a_n \)[/tex] is the leading coefficient and [tex]\( x^n \)[/tex] is the highest power term.
3. In this polynomial, the highest power of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex] (the term [tex]\( 7 x^5 \)[/tex]), making the polynomial a 5th degree polynomial.
4. According to the fundamental theorem of algebra, a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] solutions over the complex numbers.
Therefore, the number of solutions (roots) over the complex number system for the given polynomial
[tex]\[ 7 x^5 - 33 x^4 - 4 x^2 + 3 x + 52 = 0 \][/tex]
is:
[tex]\[ \boxed{5}. \][/tex]
