Find all zeros of f(x) = x³ + 5x² – 8x + 2.

Verified step by step guidance

1. First, use the Rational Root Theorem to list all possible rational roots of the polynomial. The possible rational roots are the factors of the constant term (2) divided by the factors of the leading coefficient (1).
2. Test each possible rational root by substituting them into the polynomial \( f(x) = x^3 + 5x^2 - 8x + 2 \) to see if they yield zero.
3. If a rational root is found, use polynomial division (synthetic or long division) to divide the polynomial by \( x - \text{(found root)} \) to reduce the polynomial to a quadratic.
4. Solve the resulting quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the remaining roots.
5. Combine all found roots (rational and those from the quadratic) to list all zeros of the polynomial.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = x³ + 5x² – 8x + 2 is a cubic polynomial, which means it has a degree of three. Understanding the structure of polynomial functions is essential for analyzing their behavior and finding their zeros.

Finding Zeros

Finding the zeros of a polynomial function involves determining the values of x for which f(x) = 0. These values are also known as the roots of the polynomial. Techniques for finding zeros include factoring, using the Rational Root Theorem, synthetic division, and applying numerical methods or graphing to approximate solutions.

The Rational Root Theorem

The Rational Root Theorem provides a way to identify possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem is particularly useful for narrowing down potential candidates for zeros before testing them in the polynomial.