Polynomial Calculator
- Derivative: 10x⁴ + 20x³ + 12x² + 14x − 16
- Integral: (1/3)x⁶ + x⁵ + x⁴ + (7/3)x³ − 8x² − 7x + C
- Multiply every term of P₁ by P₂, then add the partial products.
- (2x³) × (x² + 3x + 1) = 2x⁵ + 6x⁴ + 2x³.
- (−x²) × (x² + 3x + 1) = −x⁴ − 3x³ − x².
- (5x) × (x² + 3x + 1) = 5x³ + 15x² + 5x.
- (−7) × (x² + 3x + 1) = −7x² − 21x − 7.
- Combine like terms: 2x⁵ + 5x⁴ + 4x³ + 7x² − 16x − 7.
- At x = 2: P₁ × P₂ = 15 × 11 = 165. The result gives 165.
- P₁(2) = 15 and P₂(2) = 11.
Polynomial Calculator: Add, Subtract, Multiply and Divide Polynomials
This free polynomial calculator adds, subtracts, multiplies and divides two polynomials and shows the working. Type each polynomial the way you would write it, pick an operation, and the result appears as a simplified polynomial with its degree, leading coefficient, derivative, integral and a graph. Fractions are kept exact, so there is no rounding along the way.
Quick answer: multiplying 2x³ − x² + 5x − 7 by x² + 3x + 1 gives 2x⁵ + 5x⁴ + 4x³ + 7x² − 16x − 7. The calculator reaches this by multiplying each term of the first polynomial by the whole of the second and combining like terms.
What Is a Polynomial?
A polynomial is a sum of terms, where each term is a number times a variable raised to a whole-number power, such as 5x³ or −7. The highest power is the degree of the polynomial, and the number in front of it is the leading coefficient. Polynomials of degree one and two are called linear and quadratic, and you can solve those directly with the quadratic calculator. Higher degrees show up in curve fitting, physics, engineering and finance.
How Each Operation Works
To add or subtract, line up like terms, meaning terms with the same power, and combine their coefficients. To multiply, use the distributive law: every term of one polynomial is multiplied by every term of the other, and like terms are then combined. To divide, long division is used. The leading term of what is left is divided by the leading term of the divisor, the result is multiplied back and subtracted, and this repeats until the remainder has a lower degree than the divisor. The table shows one example of each.
| Operation | Example | Result |
|---|---|---|
| Add | (3x² + 2x − 5) + (x² − 4x + 9) | 4x² − 2x + 4 |
| Subtract | (3x² + 2x − 5) − (x² − 4x + 9) | 2x² + 6x − 14 |
| Multiply | (x + 2)(x − 3) | x² − x − 6 |
| Multiply | (2x³ − x² + 5x − 7)(x² + 3x + 1) | 2x⁵ + 5x⁴ + 4x³ + 7x² − 16x − 7 |
| Divide | (x³ − 2x² + x − 5) ÷ (x − 1) | Quotient x² − x, remainder −5 |
The Remainder Theorem
When you divide a polynomial by x − a, the remainder equals the value of the polynomial at x = a. In the division example above, the polynomial at x = 1 is −5, and that is exactly the remainder. A remainder of zero means the divisor is a factor of the polynomial, which is a quick way to test possible roots.
Typing Polynomials Correctly
Write powers with a caret, such as 4x^3, and use brackets for grouped terms, such as (x+1)(x−2). The multiplication sign is optional, and decimals and simple fractions are accepted. Use one variable in both boxes. If something cannot be read, the calculator points to the box and says what to change instead of guessing. For a pair of equations in two unknowns, the linear equations calculator is the better tool.
Who Uses a Polynomial Calculator?
Students use it to check algebra homework and to learn the steps of long division. Teachers use it to produce examples and answer keys. Engineers, data analysts and programmers use it when expanding formulas or simplifying expressions. For other kinds of problems, the math solver and the fraction calculator are useful companions.
How to Use This Polynomial Calculator
Enter the two polynomials
Type each one in its box, using ^ for powers. Brackets and decimals are fine.
Choose an operation
Pick Add, Subtract, Multiply or Divide. The result updates straight away.
Set a value for x
Type a number or drag the slider to see the result at that point and mark it on the graph.
Read the result and the working
See the simplified answer, its degree, derivative, integral, a check and the steps.