Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4-8x^3+29x^2-66x+72

Verified step by step guidance

1. insert step 1: Start by using the Rational Root Theorem to list all possible rational roots of the polynomial. These are the factors of the constant term (72) divided by the factors of the leading coefficient (1).
2. insert step 2: Test the possible rational roots using synthetic division or direct substitution to find any actual roots. If a root is found, it will reduce the polynomial to a lower degree.
3. insert step 3: Once a root is found, use synthetic division to divide the polynomial by the corresponding factor (x - root) to obtain a quotient polynomial of lower degree.
4. insert step 4: Repeat the process of finding roots for the quotient polynomial. Continue this process until the polynomial is reduced to a quadratic or until all roots are found.
5. insert step 5: If a quadratic polynomial remains, use the quadratic formula to find the remaining complex roots. The quadratic formula is x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a, b, and c are the coefficients of the quadratic.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is determined by the highest power of the variable. Understanding polynomial functions is crucial for analyzing their behavior, including finding zeros, which are the values of x that make the function equal to zero.

Complex Zeros

Complex zeros are solutions to polynomial equations that may not be real numbers. They can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots, counting multiplicities, which may include complex zeros when real solutions are insufficient.

Factoring and the Rational Root Theorem

Factoring a polynomial involves expressing it as a product of simpler polynomials, which can help identify its zeros. The Rational Root Theorem provides a method to find possible rational roots of a polynomial by examining factors of the constant term and the leading coefficient. This theorem is often a starting point for finding both real and complex zeros of polynomial functions.