Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=5x^3-9x^2+28x+6

Verified step by step guidance

1. <Step 1: Identify the polynomial and its degree.> The given polynomial is \( f(x) = 5x^3 - 9x^2 + 28x + 6 \). It is a cubic polynomial, which means it has a degree of 3. Therefore, it can have up to 3 complex zeros.
2. <Step 2: Use the Rational Root Theorem to find possible rational zeros.> According to the Rational Root Theorem, any rational zero of the polynomial \( f(x) \) is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (6) and \( q \) is a factor of the leading coefficient (5). List the factors of 6: \( \pm 1, \pm 2, \pm 3, \pm 6 \) and the factors of 5: \( \pm 1, \pm 5 \). Possible rational zeros are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{3}{5}, \pm \frac{6}{5} \).
3. <Step 3: Test the possible rational zeros using synthetic division.> Use synthetic division to test each possible rational zero. Start with \( x = 1 \), \( x = -1 \), etc., until you find a zero. If a zero is found, the polynomial can be factored, and the quotient will be a quadratic polynomial.
4. <Step 4: Factor the polynomial using any found zeros.> Once a rational zero is found, factor the polynomial as \( (x - \text{zero})(\text{quadratic polynomial}) \). Solve the quadratic polynomial using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the remaining zeros.
5. <Step 5: List all zeros, including complex zeros.> After solving the quadratic equation, list all zeros of the polynomial, including any complex zeros. Remember to include any multiple zeros if they occur.

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 polynomial does not have real roots.

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, determined by the highest power of the variable, influences the number of zeros it can have. For example, a cubic polynomial like ƒ(x) = 5x^3 - 9x^2 + 28x + 6 can have up to three zeros, which may be real or complex.

Finding Zeros

Finding the zeros of a polynomial involves determining the values of x for which the polynomial equals zero. This can be achieved through various methods, including factoring, synthetic division, or applying the Rational Root Theorem. For polynomials of higher degrees, such as cubic or quartic, numerical methods or the use of the quadratic formula for derived equations may be necessary to find all zeros, including complex ones.