Enter each polynomial as its coefficients from the highest power down, pick an operation, and the result appears as a simplified polynomial with its degree. For example, multiplying 1, 1 by 1, -1 gives x^2 – 1, and evaluating 2x^3 – 6x^2 + 2x – 1 at x = 3 gives 5.
Students checking algebra homework, teachers preparing answer keys and engineers expanding a characteristic equation all need the same small set of polynomial operations done without arithmetic slips. This calculator takes two polynomials as coefficient lists, highest power first, and returns their sum, difference or product in standard form, together with the degree, the coefficient list and the count of non-zero terms. In evaluate mode it computes P(x) at a chosen value using Horner's scheme and lists every multiply-and-add stage, which is handy for checking a hand evaluation line by line. The limit: it works with one variable and real coefficients up to degree 20; it does not divide, factor or find roots, so long division and solving need a different tool.
Pick P + Q, P − Q, P × Q, or P(x) at a value. The second polynomial box only appears for the first three.
List them from the highest power down, separated by commas or spaces. Use 0 for any power that is missing, so x^3 + 1 is 1, 0, 0, 1.
For the arithmetic operations enter Q the same way; for evaluation enter the x at which P should be computed.
The result shows in standard form, with its degree and coefficients beside it and each stage of the working listed below.
Result Result: x^2 – 1
Degree: 2
Coefficients (highest power first): 1, 0, -1
Non-zero terms: 2
The cross terms +x and −x cancel, leaving the difference of squares x² − 1.
Result Result: 5
Degree: 3
Coefficients (highest power first): 2, -6, 2, -1
Non-zero terms: 4
Horner's scheme runs 2 → 0 → 2 → 5, so the polynomial equals 5 at x = 3.
Result Result: 4x^2 + 2x – 3
Degree: 2
Coefficients (highest power first): 4, 2, -3
Non-zero terms: 3
Like powers combine: 3 + 1 for x², 2 + 0 for x, and 1 − 4 for the constant.
(P ± Q)_k = p_k ± q_k; (P × Q)_k = Σ p_i · q_(k−i); P(x) = (…((a_n·x + a_(n−1))·x + a_(n−2))…)·x + a_0
p_k, q_ka_n … a_0xAddition and subtraction line the two coefficient lists up by power, padding the shorter one with leading zeros, and combine them term by term. Multiplication is a discrete convolution: every term of P meets every term of Q, and products landing on the same power are summed, so degrees add. Evaluation uses Horner's nested form, which needs only n multiplications and n additions for a degree-n polynomial and avoids computing large powers separately. Leading zeros in a result are removed, so cancelling the top term lowers the reported degree.
Write one coefficient per power from x^3 down to the constant: 4, 0, -1, 7. The zero stands for the missing x^2 term. Commas, spaces or semicolons all work as separators.
When the leading coefficients are equal, subtraction cancels the top term. For example (x + 2) − (x + 0) leaves just 2, a degree-0 constant. The calculator strips leading zeros before reporting the degree.
The degree of P × Q is the sum of the two degrees, provided neither polynomial is zero. Multiplying a cubic by a quadratic therefore gives a degree-5 polynomial with up to six coefficients.
Horner's method rewrites the polynomial in nested form so a degree-n evaluation takes n multiplications and n additions. It is faster than computing each power separately and usually loses less precision, and each intermediate value it produces is the coefficient you would get from synthetic division by (x − value).
Yes. Any real number up to one billion in size is accepted, including negatives such as -3.5 and scientific notation such as 2e3. Fractions like 1/2 must be typed as decimals, 0.5.
Each polynomial can have up to 21 coefficients, which is degree 20. A product of two degree-20 polynomials reaches degree 40, still computed exactly for integer coefficients of moderate size.