Quick Answer – What is Polynomial Evaluation & Root Finding?
Polynomial evaluation computes the value of a polynomial p(x) at a specific x. Root finding locates the values of x where p(x)=0. For cubics and quartics, analytical formulas exist but are complex; numerical methods (like Newton-Raphson) are often used to approximate all real roots.
Theory & Methods
A polynomial of degree n is p(x) = anxn + ... + a1x + a0. To find its real roots, we can scan for sign changes and refine using Newton's method. For up to degree 4, exact formulas exist but this tool uses a robust hybrid algorithm: interval scanning + Newton refinement to guarantee convergence.
Newton-Raphson: xnew = x - p(x)/p'(x)
The fundamental theorem of algebra guarantees n complex roots (counting multiplicities), but real roots are the focus here. The tool also evaluates the polynomial and its derivative at any point.
Solved Examples
Example 1: Evaluate p(x)=x³-6x²+11x-6 at x=2
p(2) = 2³ - 6(2²) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0
So x=2 is a root.
Example 2: Find roots of x³-6x²+11x-6
Roots found: x=1, x=2, x=3 (all real)
Example 3: Quartic: x⁴-10x³+35x²-50x+24
Roots: x=1,2,3,4
Frequently Asked Questions
Q1. How do I enter coefficients?
Enter the coefficients in descending order of degree, from an to a0. You can use decimals, negatives, and scientific notation (e.g., 1e-3). Missing leading coefficients are treated as 0.
Q2. What if the polynomial has no real roots?
The root finder will report no real roots found. For example, x²+1 has no real roots. The tool will attempt to find any real solutions; if none exist, it says so.
Q3. Does it show complex roots?
This version is designed for real roots only. Complex roots are not displayed, but the fundamental theorem assures they exist in conjugate pairs.
Q4. How accurate are the roots?
The Newton refinement ensures at least 10-6 accuracy. The interface displays rounded values (6 decimal places) for clarity.
Q5. Can I evaluate at fractional values?
Yes, enter any real number, e.g., 2.5 or -1.333.