Final answer:
The number of complex zeros for the function f(x)=x⁴+5x²+6 is four. This is determined by expressing the polynomial as quadratics in x² and calculating the roots, which are all complex because they involve the square root of a negative number.
Step-by-step explanation:
The student is asking about the number of complex zeros for the polynomial function f(x)=x⁴+5x²+6. To find the number of complex zeros, we can factor this polynomial or use the quadratic formula. First, we recognize that the polynomial is already in a quadratic form, with a substitution of y for x²; we rewrite it as y² + 5y + 6, where y = x². This can be factored into (y + 2)(y + 3).
Next, we substitute back x² for y to get two separate quadratic equations: x² + 2 = 0 and x² + 3 = 0. Solving these equations using the quadratic formula, or by extracting square roots, gives us two complex zeros from each of the quadratics, since both 2 and 3 are positive and we would be taking the square root of a negative number. Altogether, this yields four complex zeros. The general principle from complex analysis tells us that a polynomial of degree n will have exactly n roots, considering multiplicity, in the complex number system. Since we have a fourth-degree polynomial here, we expect four zeros, and indeed that's what we find.