Polynomial Calculator
Free Polynomial Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Formula
f(x) = ax^3 + bx^2 + cx + d
Enter the coefficients a, b, c, and d for a polynomial up to degree 3. The calculator evaluates the polynomial at a given x value, computes first and second derivatives, finds roots using rational root testing, and applies Vieta's formulas for sum and product of roots.
Worked Examples
Example 1: Evaluating a Cubic Polynomial
Problem: Evaluate f(x) = x^3 - 6x^2 + 11x - 6 at x = 2, and find its roots.
Solution: f(2) = (2)^3 - 6(2)^2 + 11(2) - 6 = 8 - 24 + 22 - 6 = 0\nSince f(2) = 0, x = 2 is a root.\nUsing rational root theorem and testing: f(1) = 1 - 6 + 11 - 6 = 0, f(3) = 27 - 54 + 33 - 6 = 0\nSo the polynomial factors as (x - 1)(x - 2)(x - 3)\nDerivative: 3x^2 - 12x + 11
Result: Roots: x = 1, 2, 3 | f(2) = 0 | Degree: 3
Example 2: Finding Derivative and Critical Points
Problem: For f(x) = 2x^3 - 3x^2 - 12x + 5, find the derivative and evaluate at x = -1.
Solution: f(-1) = 2(-1) - 3(1) - 12(-1) + 5 = -2 - 3 + 12 + 5 = 12\nDerivative: f'(x) = 6x^2 - 6x - 12\nf'(-1) = 6(1) - 6(-1) - 12 = 6 + 6 - 12 = 0\nSince f'(-1) = 0, x = -1 is a critical point.\nf''(x) = 12x - 6, f''(-1) = -18 < 0, so x = -1 is a local maximum.
Result: f(-1) = 12 (local maximum) | f'(-1) = 0 | f''(-1) = -18
Frequently Asked Questions
What is a polynomial and what are its key components?
A polynomial is a mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variable exponents are non-negative integers. The key components include terms (each monomial like 3x^2), coefficients (the numerical multipliers like 3), the degree (the highest power of the variable), and the leading coefficient (the coefficient of the highest-degree term). For example, in 2x^3 - 5x^2 + 3x - 7, the degree is 3, the leading coefficient is 2, and there are four terms. Polynomials are classified by degree as linear (1), quadratic (2), cubic (3), quartic (4), and so on.
How does polynomial evaluation work?
Polynomial evaluation means computing the value of the polynomial at a specific value of x by substituting that value and performing the arithmetic. For the polynomial f(x) = 2x^3 - 5x^2 + 3x - 7, evaluating at x = 3 gives f(3) = 2(27) - 5(9) + 3(3) - 7 = 54 - 45 + 9 - 7 = 11. Horner's method provides an efficient way to evaluate polynomials by restructuring the computation as nested multiplications, reducing the number of operations. Polynomial Calculator computes the polynomial value directly using the coefficient inputs and the specified x value, giving you instant results for any combination.
What are the roots of a polynomial and how are they found?
The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero. Finding roots is one of the most important problems in algebra. For linear polynomials, there is always exactly one root. For quadratics, the quadratic formula provides exact solutions. For cubics and higher, methods include the rational root theorem, synthetic division, factoring, and numerical methods like Newton-Raphson. The Fundamental Theorem of Algebra guarantees that a polynomial of degree n has exactly n roots (counting multiplicity and complex roots). Polynomial Calculator attempts to find real roots using rational root testing and numerical approximation.
How do you find the derivative of a polynomial?
The derivative of a polynomial is found by applying the power rule to each term: the derivative of ax^n is n*a*x^(n-1). For example, the derivative of 3x^4 - 2x^3 + 5x^2 - x + 7 is 12x^3 - 6x^2 + 10x - 1. The constant term always disappears because its derivative is zero. The derivative tells you the instantaneous rate of change of the polynomial at any point and is essential for finding local maxima, minima, and inflection points. Polynomial Calculator automatically computes both the first and second derivatives, helping you analyze the behavior of your polynomial function.
What is the difference between polynomial division and synthetic division?
Polynomial long division works similarly to numerical long division, dividing one polynomial by another to obtain a quotient and remainder. It handles any polynomial divisor but can be tedious. Synthetic division is a streamlined shortcut that only works when dividing by a linear factor of the form (x - r). It uses just the coefficients and is much faster. For example, dividing x^3 - 6x^2 + 11x - 6 by (x - 1) using synthetic division takes only a few steps to show the quotient is x^2 - 5x + 6 with remainder 0, confirming that x = 1 is a root. Both methods are essential tools for factoring polynomials.
How do you factor a cubic polynomial?
Factoring a cubic polynomial typically starts with finding one rational root using the Rational Root Theorem, which says possible rational roots are factors of the constant term divided by factors of the leading coefficient. Once you find a root r, divide the cubic by (x - r) using synthetic division to get a quadratic, then factor or use the quadratic formula on the remaining quadratic. For example, x^3 - 6x^2 + 11x - 6 has possible rational roots of plus or minus 1, 2, 3, 6. Testing x = 1 gives 1 - 6 + 11 - 6 = 0, so (x - 1) is a factor. Dividing yields x^2 - 5x + 6 = (x - 2)(x - 3).