Polynomial Calculator
This polynomial calculator adds, subtracts, multiplies and divides two polynomials, and shows the working for the two that need it — a distribution grid for multiplying, the full long-division layout for dividing. All four operations look like separate skills, but one rule runs through them: powers must match before anything can be combined.
Adding and Subtracting Polynomials
Only like terms combine — terms carrying the same power. Add their coefficients and leave the power alone; x² and x³ never merge.
Subtraction has one trap that catches almost everyone: the minus applies to every term in the second polynomial, not just the first.
(3x² + 4x − 5) − (x² − 2x + 3)
= 3x² + 4x − 5 − x² + 2x − 3
= 2x² + 6x − 8
Watch the middle term: −(−2x) became +2x. Miss it and you get 2x instead of 6x — which is why the calculator flags this on every subtraction.
If that one line is what keeps catching you, it now has a page of its own — the adding and subtracting polynomials calculator sets the two out in columns and prints the second one again with every sign already changed.
Multiplying Polynomials
Every term of the first multiplies every term of the second, then like terms are collected. Coefficients multiply; exponents add. Take (2x² + 3x + 1)(x − 2) — three terms against two makes six products, and a grid keeps them all in view:
| × | x | −2 |
|---|---|---|
| 2x² | 2x³ | −4x² |
| 3x | 3x² | −6x |
| 1 | x | −2 |
Collecting: 2x³ + (−4x² + 3x²) + (−6x + x) − 2 = 2x³ − x² − 5x − 2. Degree 2 times degree 1 gave degree 3, which is always how it works.
FOIL — First, Outer, Inner, Last — is the same idea, but it only covers two binomials. The grid handles any sizes, which is why the calculator draws one.
If it is only the multiplying you are working on, the multiplying polynomials calculator names FOIL when both brackets are binomials and draws the grid when they are not.
Dividing Polynomials by Long Division
Division takes real work, and follows one loop every time: divide, multiply, subtract, repeat. Divide x³ − 2x² − 4 by x − 3:
1. x³ ÷ x = x². Multiply: x²(x − 3) = x³ − 3x². Subtract, leaving x² − 4.
2. x² ÷ x = x. Multiply: x(x − 3) = x² − 3x. Subtract, leaving 3x − 4.
3. 3x ÷ x = 3. Multiply: 3(x − 3) = 3x − 9. Subtract, leaving 5.
The quotient is x² + x + 3, remainder 5. We stop because 5 has a lower degree than x − 3.
Check it the same way every time — divisor × quotient + remainder must return the original: (x − 3)(x² + x + 3) + 5 = x³ − 2x² − 4. When the remainder is 0 the divisor is a factor: (6x² + x − 2) ÷ (2x − 1) gives 3x + 2 exactly.
How to Use This Polynomial Calculator
Type both polynomials and pick the operation. Terms can be in any order, brackets like (x+1)² are expanded for you, and fractional coefficients are fine. Tap +, −, × or ÷ and the answer updates at once.
The keypad serves whichever box you tapped last, and that box is highlighted. AC clears only the active box, and each field has its own keyboard toggle.
Read the band, then the working. The answer appears in standard form with a remainder chip when division leaves one. Below it: a distribution grid for multiplication, or the full long-division layout for division — quotient above the bar, each subtraction ruled off, the remainder marked. "Show the working" opens the products before collecting, or the divisor × quotient + remainder check.
What This Calculator Does Differently
All four polynomial operations in one tool. Several sites split adding, multiplying and dividing across three separate calculators, so you change pages to change operation. Here it is one button.
The working is drawn, not described. Every product gets its own cell in the grid; the long division is laid out the way you would write it on paper. Most free tools give the answer and a line of text.
Exact fractions, never decimals. Divide x² + 1 by 2x − 1 and the quotient is exactly ½x + ¼ with remainder &frac54; — nothing is rounded. It also warns where marks are lost: the sign flip on subtraction, a zero remainder meaning the divisor is a factor, and a dividend too small to divide at all.
Worked Examples
| Problem | Answer |
|---|---|
| (3x² + 4x − 5) − (x² − 2x + 3) | 2x² + 6x − 8 |
| (2x² + 3x + 1)(x − 2) | 2x³ − x² − 5x − 2 |
| (x + 3)(x − 3) | x² − 9 |
| (x³ − 2x² − 4) ÷ (x − 3) | x² + x + 3, remainder 5 |
| (2x² − 5x − 3) ÷ (x − 3) | 2x + 1 — divides exactly |
| (x² + 1) ÷ (2x − 1) | ½x + ¼, remainder &frac54; |
Frequently Asked Questions
How do you add and subtract polynomials?
How do you multiply polynomials?
What is FOIL and when can I use it?
How do you divide polynomials by long division?
What is the remainder in polynomial division?
Can this calculator divide by a binomial like 2x − 1?
What is the degree of the answer?
Is this polynomial calculator free?
Four polynomial operations, one rule: powers must match before terms combine. Add and subtract by collecting like terms, multiply by pairing every term with every term, and divide by the divide-multiply-subtract loop until the leftover is too small to continue. The calculator above runs all four in exact arithmetic and draws both the grid and the long division. When it is a single polynomial you need to analyse rather than combine, the remainder theorem calculator handles linear divisors in one line, and the rest of the free toolkit is on the Monkza calculators page.