Add, subtract, multiply, and divide polynomials with complete step-by-step solutions. Supports single and multi-variable polynomial operations.
Step 1: Identify like terms: x² terms (3x² + x²), x terms (2x - 4x), constants (-1 + 7)
Step 2: Combine coefficients: 3 + 1 = 4 for x², 2 - 4 = -2 for x, -1 + 7 = 6
Result: 4x² - 2x + 6
Step 1: Distribute the minus sign: 5x³ - 3x² + 2x - 2x³ - x² + 5
Step 2: Combine like terms: (5-2)x³ = 3x³, (-3-1)x² = -4x², 2x, +5
Result: 3x³ - 4x² + 2x + 5
Step 1: Use FOIL method: (2x)(x) + (2x)(-4) + (3)(x) + (3)(-4)
Step 2: Simplify: 2x² - 8x + 3x - 12
Step 3: Combine like terms: 2x² - 5x - 12
Result: 2x² - 5x - 12
Step 1: Divide the leading term: x² ÷ x = x
Step 2: Multiply: x(x + 2) = x² + 2x. Subtract: (x² + 5x) - (x² + 2x) = 3x
Step 3: Bring down +6. Divide: 3x ÷ x = 3. Multiply: 3(x + 2) = 3x + 6. Subtract: (3x + 6) - (3x + 6) = 0
Result: x + 3 (remainder 0)
A polynomial is an expression consisting of variables (also called indeterminates), coefficients, and exponents that are combined using addition, subtraction, multiplication, and non-negative integer exponents. For example, 3x² + 2x - 1 is a polynomial in variable x.
To add or subtract polynomials, identify like terms (terms with the same variable raised to the same power) and combine their coefficients. For subtraction, distribute the negative sign across all terms of the second polynomial before combining.
Multiply polynomials using the distributive property (often called FOIL for binomials: First, Outer, Inner, Last). Each term in the first polynomial must be multiplied by each term in the second polynomial.
Polynomial long division is similar to numerical long division. Divide the leading term of the dividend by the leading term of the divisor, multiply the result by the entire divisor, subtract from the dividend, and repeat until the degree of the remainder is less than the degree of the divisor.
Terms with the same variable(s) raised to the same exponent(s). Example: 3x² and -2x² are like terms; 3x² and 3x³ are not.
The highest exponent of any term. A polynomial of degree n has at most n+1 terms when written in standard form.
The coefficient of the term with the highest degree. In 3x² + 2x - 1, the leading coefficient is 3.
Terms arranged in descending order of degree. Example: 4x³ + 2x² - x + 7 (not 2x² + 7 + 4x³ - x).
A polynomial is a mathematical expression consisting of variables (also called indeterminates), coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication. Polynomials are fundamental to algebra and appear throughout mathematics, science, and engineering.
The general form of a polynomial in one variable is: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ, aₙ₋₁, ..., a₀ are constants (coefficients) and n is a non-negative integer representing the degree of the polynomial. For example, 4x³ - 2x² + 5x - 7 is a polynomial of degree 3 (a cubic polynomial).
Polynomials can also have multiple variables, such as 3x²y + 2xy² - xy + 4. Each term is a product of a coefficient and variables raised to powers. The degree of a multi-variable term is the sum of the exponents of the variables in that term.
Our Polynomial Calculator makes it easy to perform arithmetic operations on polynomials. Here's how to use it:
Step 1: Select the operation you want to perform — Add, Subtract, Multiply, or Divide — using the radio buttons at the top of the calculator.
Step 2: Enter your polynomial expressions. Use the caret (^) symbol for exponents. For example, type "3x^2 + 2x - 1" for 3x² + 2x - 1. The calculator accepts polynomials with any variable name (x, y, z, etc.) and handles multi-variable polynomials.
Step 3: Click the "Calculate" button to see the result and a detailed step-by-step solution showing how the operation was performed.
The calculator automatically parses your input, identifies like terms, and provides a clear breakdown of each calculation step, making it an excellent tool for learning and verification.
x^2, y^3, 4x^53x, -2y, 5x^2y^37, -33x^2+2x-1 works the same as 3x^2 + 2x - 1Polynomials are used extensively in mathematics and its applications. Here are some common areas where polynomial operations are essential:
Polynomials model projectile motion (quadratic), describe electrical circuits, and represent stress-strain relationships in materials science.
Revenue and cost functions are often polynomial. Profit maximization, break-even analysis, and compound interest calculations use polynomial equations.
Bezier curves and splines are polynomial functions used to create smooth curves in graphic design, animation, and font rendering.
Polynomial regression fits curved relationships between variables. Polynomial interpolation is used to estimate values between known data points.
⚠️ Important Note: This Polynomial Calculator is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as engineering design, scientific research, or academic submissions. Always show your own working when required by instructors.