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

Verified step by step guidance

1. Identify the polynomial function given: $f\left(x\right)=x^3+5x^2-8x+2$. We need to find all values of $x$ such that $f\left(x\right)=\; 0$.
2. Use the Rational Root Theorem to list possible rational zeros. The possible rational roots are factors of the constant term (±1, ±2) divided by factors of the leading coefficient (±1), so possible roots are $\pm 1,\; \pm 2$.
3. Test each possible root by substituting into the polynomial or using synthetic division to check if it yields zero. When a root is found, factor it out from the polynomial.
4. After factoring out the found root, you will get a quadratic polynomial. Solve this quadratic equation using the quadratic formula $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$ to find the remaining zeros.
5. Combine all roots found from the rational root test and the quadratic solution to list all zeros of the polynomial function.

Key Concepts

Polynomial Zeros

Zeros of a polynomial are the values of x for which the polynomial equals zero. Finding zeros involves solving the equation f(x) = 0, which helps identify the roots or x-intercepts of the polynomial function.

Rational Root Theorem

The Rational Root Theorem provides possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. It helps narrow down candidates to test when searching for roots of polynomials with integer coefficients.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial, especially after finding one root. It simplifies the polynomial to a lower degree, making it easier to find remaining zeros by factoring or solving quadratic equations.