According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has exactly \( n \) roots, counting multiplicities, in the complex number system. In your case, the polynomial has a term \( 8x^5 \), meaning it is of degree 5.
From this, we can conclude that:
- The polynomial has exactly 5 roots (which could be real or complex).
Now, let's analyze each of the provided statements:
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The equation has an odd number of real roots. - This statement is not necessarily true; the polynomial could have 5 real roots, 3 real roots and 2 complex roots, 1 real root and 4 complex roots, or 5 complex roots (where complex roots come in conjugate pairs).
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The equation has at least 5 roots. - This statement is true because a degree 5 polynomial has exactly 5 roots, so "at least 5" qualifies.
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The equation has more than 5 roots. - This statement is false since a degree 5 polynomial can only have 5 roots.
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All the roots of the equation are real. - This statement is also not necessarily true; the roots can be a combination of real and complex roots.
Therefore, the correct statement that must be true is:
The equation has at least 5 roots.