Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=x^4+x^3-9x^2+11x-4$

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = x^4 + x^3 - 9x^2 + 11x - 4\).
2. Use the Rational Root Theorem to list possible rational zeros. These are factors of the constant term (\(-4\)) divided by factors of the leading coefficient (\(1\)), so possible rational roots are \(\pm1, \pm2, \pm4\).
3. Test each possible rational root by substituting into \(f(x)\) or by using synthetic division to see if it yields zero. When a root is found, factor it out from the polynomial.
4. After factoring out the linear factors corresponding to the roots found, you will get a quadratic or cubic polynomial. Solve the remaining polynomial using factoring, completing the square, or the quadratic formula to find the remaining zeros.
5. List all zeros found, including complex zeros if any arise from solving the quadratic or cubic factor, and express them in exact form.

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 Techniques

Factoring a polynomial involves expressing it as a product of lower-degree polynomials, which helps identify zeros. Techniques include synthetic division, factoring by grouping, and using the Rational Root Theorem to test possible rational zeros. Factoring simplifies finding exact roots, including complex ones.

Fundamental Theorem of Algebra

This theorem states that every non-constant polynomial with complex coefficients has at least one complex root. Consequently, a polynomial of degree n has exactly n roots in the complex number system, counting multiplicities. This guarantees that all zeros, real or complex, can be found for the given polynomial.