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Complex zeros occur when a polynomial has solutions that include imaginary numbers. Imaginary numbers are represented as multiples of "i," where "i" is the imaginary unit defined as \(i = \sqrt{-1}\). Complex zeros always come in pairs known as conjugates, which means if \(a + bi\) is a zero, then \(a - bi\) will also be a zero. For polynomial functions with real coefficients, these conjugate pairs ensure the coefficients remain real.

In the exercise, we checked if \(-3i\) and \(3i\) are zeros of the function \(f(x) = x^3 + 9x\). When substituting these into the polynomial, the outcome was not equal to zero, indicating they are not actual zeros of this function.

Understanding complex zeros is essential as they showcase the behavior of polynomials in the complex plane. Even if a polynomial has no real zeros, it can still have complex zeros, which is often the case for polynomials of odd degrees.

Evaluating a function involves substituting a specific value for the variable into the function's expression and simplifying the result. This process is crucial for determining whether a given number is a zero of the function.

• First, identify the function and the value to substitute.
• Replace the variable in the function with the given number.
• Simplify the resulting expression to find the outcome.

For example, to evaluate the function \(f(x) = x^3 + 9x\) at \(x = 0\), substitute zero for \(x\):\[f(0) = 0^3 + 9 \times 0 = 0\] Since the outcome is zero, zero is a zero of the function. In contrast, substituting \(-3i\) and \(3i\) into \(f(x)\) gave non-zero outcomes, confirming those are not zeros of the function. Mastering evaluation techniques helps in verifying solutions and understanding the roots of polynomial equations.

Zeros of polynomials, also known as roots or solutions, are values for which the polynomial equals zero. They reveal the points where the graph of the polynomial functions crosses the x-axis.

There are different types of zeros:

• Real Zeros: These are where the polynomial crosses the x-axis at real number points.
• Complex Zeros: Involve imaginary numbers and do not cross the x-axis but suggest oscillations in behavior.

For the polynomial \(f(x) = x^3 + 9x\), the zero \(x = 0\) indicates it crosses the x-axis at the origin. Other possible zeros, if any were actual zeros, would be expressed as complex numbers that denote interaction in the imaginary plane.

Zeros are vital for graphing polynomial functions and solving equations, as they provide insight into the function's behavior and intersections with axes.