The Polynomial Calculator helps you work with polynomial expressions: adding, subtracting, multiplying, dividing, factoring, and finding roots. Polynomials are fundamental algebraic expressions that appear throughout mathematics, from basic algebra to calculus and beyond.

Key Features

• Polynomial Operations: Add, subtract, multiply, and divide polynomials with step-by-step solutions.
• Factoring: Factor polynomials completely, including greatest common factors, difference of squares, and trinomials.
• Finding Roots: Calculate all real and complex zeros of polynomials using various methods.
• Long Division: Perform polynomial long division with detailed steps shown.
• Synthetic Division: Use synthetic division for faster division by linear factors.
• Evaluation: Substitute values and evaluate polynomials at specific points.
• Graphing: Visualize polynomial functions to understand their behavior and key features.

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents.

Example 1: Adding Polynomials

Add (3x² + 2x - 5) + (x² - 4x + 7):

Combine like terms:
x<sup>2</sup> terms: 3x<sup>2</sup> + x<sup>2</sup> = 4x<sup>2</sup>
x terms: 2x + (-4x) = -2x
Constants: -5 + 7 = 2

Result: 4x<sup>2</sup> - 2x + 2

Example 2: Multiplying Polynomials

Multiply (x + 3)(x - 2):

Use FOIL method:
First: x × x = x<sup>2</sup>
Outer: x × (-2) = -2x
Inner: 3 × x = 3x
Last: 3 × (-2) = -6

Combine like terms:
x<sup>2</sup> + (-2x) + 3x + (-6) = x<sup>2</sup> + x - 6

Result: x<sup>2</sup> + x - 6

Example 3: Factoring Quadratic

Factor x² + 7x + 12:

Find two numbers that:
- Multiply to 12 (constant term)
- Add to 7 (coefficient of x)

Numbers: 3 and 4
- 3 × 4 = 12 ✓
- 3 + 4 = 7 ✓

Result: (x + 3)(x + 4)

Verification:
(x + 3)(x + 4) = x<sup>2</sup> + 4x + 3x + 12 = x<sup>2</sup> + 7x + 12 ✓

Example 4: Polynomial Long Division

Divide (x³ + 2x² - 5x - 6) by (x - 2):

          x<sup>2</sup> + 4x + 3
       _______________
x - 2 | x<sup>3</sup> + 2x<sup>2</sup> - 5x - 6
        x<sup>3</sup> - 2x<sup>2</sup>
        ________
            4x<sup>2</sup> - 5x
            4x<sup>2</sup> - 8x
            ________
                 3x - 6
                 3x - 6
                 ______
                     0

Result: x<sup>2</sup> + 4x + 3 (remainder 0)
This means (x - 2) is a factor!

Example 5: Finding Roots

Find roots of x² - 5x + 6 = 0:

Method 1: Factoring
x<sup>2</sup> - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3

Method 2: Quadratic Formula
x = (-b ± √(b<sup>2</sup> - 4ac)) / (2a)
where a = 1, b = -5, c = 6

x = (5 ± √(25 - 24)) / 2
x = (5 ± 1) / 2
x = 3 or x = 2

Roots: x = 2, x = 3

• Combine Like Terms: Only terms with the same variable and exponent can be combined: 3x² + 2x² = 5x², but 3x² + 2x cannot be simplified.
• Factoring Strategy: Always look for GCF first, then check for special patterns (difference of squares, perfect square trinomials), then try factoring by grouping or trinomial factoring.
• Degree of Product: When multiplying polynomials, the degree of the result equals the sum of the degrees: (degree 2) × (degree 3) = degree 5.
• Remainder Theorem: If polynomial P(x) is divided by (x - a), the remainder is P(a). Use this to check factors quickly.
• Rational Root Theorem: Possible rational roots are factors of constant term divided by factors of leading coefficient.
• Fundamental Theorem: A polynomial of degree n has exactly n roots (counting multiplicities and complex roots).
• Synthetic Division: Faster than long division for dividing by (x - a), but only works with linear divisors.
• Sign Changes: Descartes' Rule of Signs: the number of positive real roots is at most the number of sign changes in P(x).