What it does
This calculator multiplies polynomials in x and writes the answer out in full, highest power first. It shows each product of a term from one bracket with a term from the other, then adds the like terms. Use it to check algebra homework, to expand something like (x + 1)³ before you integrate it, or to see where a coefficient in a long product comes from. Coefficients can be whole numbers, fractions or decimals, and the arithmetic is exact, so 1/3 stays 1/3 instead of becoming 0.3333.
How polynomials are multiplied
The rule is the distributive property, used over and over: multiply each term of the first polynomial by each term of the second, then add the results and combine like terms (terms with the same power of x). To multiply two terms, multiply the numbers and add the powers.
c·xm × d·xn = (c × d)·xm+n
A polynomial with m terms times one with n terms gives m × n products before combining. For two binomials, that's the four products of FOIL: First, Outer, Inner, Last. For three or more polynomials, multiply the first two, then multiply that answer by the next one.
Worked example: (2x − 1)(x² + 3x − 4)
- 2x times each term: 2x · x² = 2x³, 2x · 3x = 6x², 2x · (−4) = −8x
- −1 times each term: −1 · x² = −x², −1 · 3x = −3x, −1 · (−4) = 4
- Combine like terms: 6x² − x² = 5x², and −8x − 3x = −11x
- Answer: 2x³ + 5x² − 11x + 4
Quick check: at x = 2 the factors are 3 and 6, and 3 × 6 = 18. The answer at x = 2 is 16 + 20 − 22 + 4 = 18, so it matches. The calculator does this check for you.
Patterns worth knowing
| Product | Expanded | Name |
|---|---|---|
| (a + b)² | a² + 2ab + b² | Perfect square |
| (a − b)² | a² − 2ab + b² | Perfect square |
| (a + b)(a − b) | a² − b² | Difference of squares |
| (x + 1)³ | x³ + 3x² + 3x + 1 | Binomial cube |
Limits
- One variable only, written x. Expressions like (x + y)² need a multivariable tool.
- Powers must be whole numbers 0 or more, so x−1 or √x aren't polynomials and are rejected.
- You can divide by a number (x/2) but not by an expression containing x, since that isn't a polynomial.
- The answer can have powers up to x100. Very long products skip the term-by-term list and show the totals.