Polynomial calculator
Type a polynomial expression: add, subtract, multiply or divide, in exact arithmetic with every step shown.
Use ^ for powers and parentheses for products: (x+1)(x−1), 2x(x+3), (x+2)^3. Any single letter works as the variable.
How do I add and subtract polynomials?
Combine like terms, the terms with the same power. For example, (12x² + 9x − 21) + (4x³ + 8x² − 5x + 20) = 4x³ + 20x² + 4x − 1, and (7x⁴ − x² + 6x + 1) − (5x³ − 2x² + 3x + 2) = 7x⁴ − 5x³ + x² + 3x − 1 (OpenStax Algebra and Trigonometry 2e, section 1.4, Examples 2 and 3). When subtracting, change the sign of every term in the second polynomial first.
How do I multiply polynomials, and what is FOIL?
Multiply every term of one polynomial by every term of the other, then combine like terms: (2x + 1)(3x² − x + 4) = 6x³ + x² + 7x + 4. For two binomials the four products are First, Outer, Inner and Last: (2x − 18)(3x + 3) = 6x² + 6x − 54x − 54 = 6x² − 48x − 54. Two special cases are (3x − 8)² = 9x² − 48x + 64 and the difference of squares (9x + 4)(9x − 4) = 81x² − 16 (OpenStax, section 1.4, Examples 4 to 7). Enter two binomials and the calculator shows the four FOIL steps.
How does polynomial long division work?
Divide the leading term of the dividend by the leading term of the divisor, multiply that quotient term by the whole divisor, subtract, and repeat with what is left until its degree is below the divisor’s. That final piece is the remainder. (6x³ + 11x² − 31x + 15) ÷ (3x − 2) gives the quotient 2x² + 5x − 7 and remainder 1, written 2x² + 5x − 7 + 1/(3x − 2) (OpenStax, section 5.4, Example 2). Check: (3x − 2)(2x² + 5x − 7) + 1 equals the dividend.
What is synthetic division, and how does the remainder relate to it?
A shortcut for dividing by x − c that uses only the coefficients. For (−9x⁴ + 10x³ + 7x² − 6) ÷ (x − 1), the bottom row is −9, 1, 8, 8, 2, so the quotient is −9x³ + x² + 8x + 8 with remainder 2 (OpenStax, section 5.4, Example 5). By the remainder theorem the remainder is the value of the dividend at x = c: here the dividend at x = 1 is 2. When the divisor has the form x − c with whole-number coefficients, the Divide mode shows that bottom row; the synthetic division calculator has the full table.
What are the degree and leading coefficient?
Write the polynomial in standard form, with the powers descending. The degree is the highest power and the leading coefficient is the number in front of that term. 3 + 2x² − 4x³ becomes −4x³ + 2x² + 3, so its degree is 3 and its leading coefficient is −4 (OpenStax, section 1.4, Example 1).
How do I type a polynomial, and what is supported?
Use ^ for powers (x^2 or x²), + and − for sums, * or just parentheses for products: 2x(x + 1), (x + 1)(x − 1). Whole-number powers 0 and up, decimals such as 0.25 and fractions such as 1/3 work, and any single letter can be the variable. Arithmetic is exact (big integers and fractions), so (x + 1)^20 shows every digit. Division by a number is fine inside an expression; to divide by a polynomial use the Divide mode. The degree is limited to 200.
How do I find the roots?
A polynomial of degree 1 has one root; for degree 2 the calculator gives the roots from the quadratic formula, as exact fractions and radicals when the coefficients allow, and shows complex roots with i. For the discriminant and step-by-step work use the quadratic formula calculator. Degree 3 has its own cubic equation calculator; for higher degrees, test a possible root c by dividing by (x − c): the remainder is zero exactly when c is a root.
Sources: OpenStax, Algebra and Trigonometry 2e, section 1.4 “Polynomials” (Examples 1 to 7) and section 5.4 “Dividing Polynomials” (Examples 1 to 5). Related: quadratic formula calculator, synthetic division calculator, cubic equation calculator, factor calculator and completing the square calculator.