Final answer:
To find the zeros of the polynomial f(x)=x⁴-x-3x²+x+2, one might need to employ methods such as factoring or the Rational Root Theorem, followed by the use of the quadratic formula for any resulting quadratic factors.
Step-by-step explanation:
The goal is to find the four real zeros of the polynomial f(x)=x⁴-x³-3x²+x+2. To do this, we may need to use various methods such as factoring, the Rational Root Theorem, synthetic division, or finding complex roots and their conjugates. Polynomials of the fourth degree do not generally have simple formulas for solving, unlike second-degree polynomials, which can be solved using the quadratic formula.
For quadratic equations of the form ax² + bx + c = 0, you can find the solutions for x using the quadratic formula:
x = √(b² - 4ac) - b)/(2a), where a, b, and c are coefficients of the equation. To apply this method to finding the zeros of f(x), one would typically have to reduce the fourth-degree polynomial to a quadratic equation, which is not straightforward in this case. However, if a polynomial is factored into quadratic factors or we find some real zeros, then we can use the quadratic formula to solve for the remaining zeros.