Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $\text{ƒ}\left(x\right)=2x^5+11x^4+16x^3+15x^2+36x$

Verified step by step guidance

1. First, factor out the greatest common factor (GCF) from the polynomial ƒ(x) = 2x^5 + 11x^4 + 16x^3 + 15x^2 + 36x. Identify the GCF of all terms and factor it out.
2. After factoring out the GCF, you will have a polynomial of degree 4 inside the parentheses. Next, attempt to factor this quartic polynomial further by using methods such as factoring by grouping or synthetic division to find at least one root.
3. Use the Rational Root Theorem to list possible rational roots of the quartic polynomial. Test these possible roots by substitution or synthetic division to find actual roots.
4. Once a root is found, use synthetic division or polynomial division to divide the quartic polynomial by the corresponding factor (x - root), reducing the polynomial's degree.
5. Repeat the process of finding roots and factoring until the polynomial is completely factored into linear and/or irreducible quadratic factors. Then, solve each factor equal to zero to find all complex zeros of the original polynomial.

Key Concepts

Polynomial Zeros and Roots

Zeros or roots of a polynomial are the values of x for which the polynomial equals zero. Finding these involves solving the equation ƒ(x) = 0. Complex zeros include real and non-real (imaginary) solutions, and the Fundamental Theorem of Algebra guarantees that a polynomial of degree n has exactly n roots in the complex number system.

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of simpler polynomials. It helps in finding zeros by setting each factor equal to zero. Techniques include factoring out the greatest common factor, grouping, and using special formulas, which simplify solving higher-degree polynomials like the given fifth-degree function.

Complex Numbers and Imaginary Units

Complex numbers have a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with i² = -1. When polynomial equations have no real solutions, complex solutions arise. Understanding how to work with complex numbers is essential for expressing all zeros of polynomials, especially when factoring leads to quadratic factors with negative discriminants.