Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁴+2x³-3x²+24x-180

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = x^4 + 2x^3 - 3x^2 + 24x - 180\).
2. Look for possible rational zeros using the Rational Root Theorem. The possible rational zeros are factors of the constant term (\(-180\)) divided by factors of the leading coefficient (which is 1). So, possible rational zeros are \(\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 9, \pm 10, \pm 15, \pm 18, \pm 30, \pm 45, \pm 60, \pm 90, \pm 180\).
3. Test these possible rational zeros by substituting them into the polynomial or by using synthetic division to find which values make the polynomial equal to zero. Each successful root will help factor the polynomial.
4. Once a root \(r\) is found, factor out \((x - r)\) from the polynomial using polynomial division or synthetic division to reduce the polynomial to a cubic or quadratic.
5. Repeat the process of finding zeros for the reduced polynomial until it is factored completely into linear and/or quadratic factors. Then solve the quadratic factors using the quadratic formula if necessary to find all complex zeros.

Key Concepts

Complex Zeros of Polynomial Functions

Complex zeros are the values of x, including real and non-real complex 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 Division and Factoring

Factoring polynomials or using polynomial division (such as synthetic or long division) helps break down higher-degree polynomials into simpler factors. This process is essential to identify zeros by reducing the polynomial to linear or quadratic factors, which can then be solved more easily.

Quadratic Formula and Solving Quadratics

When a polynomial is factored into quadratic expressions that cannot be factored further, the quadratic formula is used to find the roots. The formula x = (-b ± √(b² - 4ac)) / 2a provides exact solutions, including complex roots when the discriminant is negative.