Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴-8x³+29x²-66x+72

Verified step by step guidance

1. Start by examining the polynomial function \(f(x) = x^4 - 8x^3 + 29x^2 - 66x + 72\) to find its complex zeros. Since it is a quartic (degree 4), there will be 4 zeros in total, counting multiplicities and complex roots.
2. Attempt to find rational roots using the Rational Root Theorem. The possible rational roots are factors of the constant term 72 divided by factors of the leading coefficient 1, so test \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm9, \pm12, \pm18, \pm24, \pm36, \pm72\) by substituting into \(f(x)\).
3. Once a root \(r\) is found, use polynomial division (either synthetic or long division) to divide \(f(x)\) by \((x - r)\), reducing the quartic to a cubic polynomial.
4. Repeat the process of finding roots and factoring the polynomial until it is factored completely into linear and/or quadratic factors. If a quadratic factor remains that cannot be factored further over the reals, use the quadratic formula to find its complex zeros.
5. Write down all zeros found from the linear factors and the solutions from the quadratic formula, ensuring to express the complex zeros in exact form (using \(i\) for the imaginary unit).

Key Concepts

Complex Zeros of Polynomial Functions

Complex zeros are the values of x, possibly including imaginary numbers, that make the polynomial equal to zero. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex zeros, counting multiplicities. Finding these zeros involves solving the polynomial equation ƒ(x) = 0.

Polynomial Factoring and Synthetic Division

Factoring polynomials or using synthetic division helps break down higher-degree polynomials into simpler factors. This process is essential to find zeros by reducing the polynomial to linear or quadratic factors, which can then be solved directly or by applying the quadratic formula.

Quadratic Formula for Solving Quadratic Equations

The quadratic formula, x = (-b ± √(b² - 4ac)) / 2a, is used to find exact roots of quadratic equations. When factoring leads to quadratic factors that cannot be factored further, this formula provides the exact complex or real zeros, including those with imaginary parts.