Polynomial Roots Calculator

About Polynomial Roots Calculator

Polynomial roots are the x-values where the polynomial equals zero. Quadratic uses the quadratic formula; higher degrees use numerical methods.

How to use this calculator

1. Enter the polynomial coefficients from the highest power down to the constant term.
2. Make sure the leading coefficient is not 0, because that would change the degree.
3. Check the degree, this tool handles polynomials up to degree 4.
4. Read the roots, which may be real numbers, repeated roots, or complex numbers.

The formula explained

A polynomial has the form \(p(x) = a_nx^n + \cdots + a_0\), where the roots are the values of \(x\) that make \(p(x)=0\). This calculator finds those roots for polynomials of degree 4 or lower.

• x = the variable in the polynomial
• n = the degree of the polynomial, up to 4
• \(a_n\) = the leading coefficient of the highest-power term
• \(a_0\) = the constant term

Step by step method

1. Write the polynomial in standard form, with powers of x in descending order.
2. Set the polynomial equal to 0 and look for values of x that satisfy the equation.
3. For simple cases, factor the polynomial or use known formulas for quadratic, cubic, or quartic equations.
4. Check your answers by substituting each root back into the polynomial.

Worked example

Problem. Find the roots of \(x^2 - 5x + 6 = 0\).

1. Factor the quadratic: \(x^2 - 5x + 6 = (x - 2)(x - 3)\).
2. Set each factor equal to 0: \(x - 2 = 0\) or \(x - 3 = 0\).
3. Solve to get \(x = 2\) and \(x = 3\).

Tips and common mistakes

• Always include every power of x in order, even if a coefficient is 0.
• A repeated root can appear more than once, and some degree 4 polynomials have complex roots.

Frequently asked questions