Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x^4+x^3-9x^2+11x-4

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 (-4) divided by the factors of the leading coefficient (1).
2. insert step 2: Test each possible rational root by substituting them into the polynomial \( f(x) = x^4 + x^3 - 9x^2 + 11x - 4 \) to see if they yield zero.
3. insert step 3: Once a root is found, use polynomial division (synthetic or long division) to divide the polynomial by \( x - \text{(found root)} \) to reduce the polynomial's degree.
4. insert step 4: Repeat the process of finding roots and dividing the polynomial until you reduce it to a quadratic equation.
5. insert step 5: Solve the resulting quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the remaining complex roots.

Key Concepts

Complex Zeros

Complex zeros are the solutions to a polynomial equation that may include real and imaginary numbers. A complex zero can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. Understanding complex zeros is essential for analyzing polynomial functions, especially when the degree of the polynomial is higher than two, as they may not always yield real solutions.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable x is given by f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer and a_n is not zero. Understanding the structure of polynomial functions is crucial for finding their zeros and analyzing their behavior.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of simpler polynomials or linear factors. This process is vital for finding the zeros of the polynomial, as the zeros correspond to the values of x that make each factor equal to zero. Techniques such as synthetic division, the Rational Root Theorem, and grouping can be employed to factor polynomials effectively.