Multiplying Polynomials Calculator
FOIL is the most useful thing a student learns about multiplying brackets, and it is also the thing that causes the most damage — because nobody tells them where it stops.
First, Outer, Inner, Last works beautifully on (2x + 3)(x − 5). It does not work on (x² + 2x + 1)(x − 1), and the reason is arithmetic rather than opinion: FOIL names four products, and that multiplication has six. Two of them have nowhere to go. This multiplying polynomials calculator names FOIL when it genuinely applies and shows a grid when it does not, so you can see which case you are in before you start.
Count the Products First
Before multiplying anything, count. Terms in the first bracket times terms in the second is how many products you will end up with:
| binomial × binomial | 2 × 2 = 4 — FOIL |
| trinomial × binomial | 3 × 2 = 6 — grid |
| trinomial × trinomial | 3 × 3 = 9 — grid |
That number is your safety net. Work the multiplication, count what you wrote down, and if the two do not match you have dropped one. It takes three seconds and it catches most errors before they reach the answer.
FOIL, and Exactly Where It Ends
Take (2x + 3)(x − 5):
First 2x × x = 2x²
Outer 2x × −5 = −10x
Inner 3 × x = 3x
Last 3 × −5 = −15
Four products, and the middle two are both x terms, so they collect: −10x + 3x = −7x. The answer is 2x² − 7x − 15.
FOIL is not a method. It is a mnemonic for the order you take four products in, and the four exist only because 2 × 2 = 4. The moment either bracket has a third term, the letters run out. There is no FOILL, and inventing one is how terms go missing.
The Grid Does What FOIL Cannot
For (3x² − x + 2)(2x² + 4x − 1), draw a box: one row per term on the left, one column per term on the top, and multiply into each cell.
Nine cells, nine products, and nothing to remember — you simply cannot skip one, because an empty cell is visible. Then look down the grid for cells that share a power and add those together.
This is the same thing FOIL does, drawn out. Once you have seen the 2×2 grid you realise FOIL was always just the four cells of a small box read in a particular order.
Collecting Like Terms Is Where Marks Go
Almost nobody gets the multiplying wrong. What goes wrong is the adding afterwards.
Only terms with the same power can be combined. 6x³ and 4x³ add to 10x³; 6x³ and 4x² do not add at all. In a nine-cell grid there may be three or four powers that appear more than once, and each is a separate small addition with its own chance of a sign error.
The calculator marks the cells that will combine in green and then shows each collection on its own line, so the multiplying and the adding stay visibly separate.
The Degree Check
The degree of a product is the two degrees added. Multiply a cubic by a quadratic and the answer is degree 5, always — because the two leading terms multiply and nothing else can reach that high.
So glance at your answer's highest power before you write it down. Degree 4 when it should be 5 means a term vanished. It is the fastest check in algebra and it takes no working at all.
How to Use This Multiplying Polynomials Calculator
Put a polynomial in each box and press =. Either can have any number of terms, in any order, and brackets are expanded first. Tap a box to aim the keypad at it; the key icon beside a box switches that one to your own keyboard.
The band gives the product and says how many products were made and how many needed collecting. Tap the button below it and the working opens — FOIL laid out by letter if both are binomials, the full grid if not, with the cells that combine in green. Underneath, the steps: every term times every term, the collection, and the degree check at the end.
If you want the reverse of this — taking a product apart rather than building one — the factoring polynomials calculator does that, and multiplying back out is exactly how you check a factorisation.
Worked Examples
| Multiplication | Products | Answer |
|---|---|---|
| (x + 2)(x + 3) | 4 — FOIL | x² + 5x + 6 |
| (2x + 3)(x − 5) | 4 — FOIL | 2x² − 7x − 15 |
| (x − 1)(x − 1) | 4 — FOIL | x² − 2x + 1 |
| (x² + 2x + 1)(x − 1) | 6 — grid | x³ + x² − x − 1 |
| (3x² − x + 2)(2x² + 4x − 1) | 9 — grid | 6x⁴ + 10x³ − 3x² + 9x − 2 |
Row three is worth trying by hand. Squaring a bracket is still four products, not two — (x − 1)² is not x² − 1, and the middle term is the one everyone forgets.
Frequently Asked Questions
How do you multiply polynomials?
What is the FOIL method?
Can you use FOIL for trinomials?
What is the degree of the product of two polynomials?
Why do you collect like terms after multiplying?
How many products are there when multiplying polynomials?
Is this multiplying polynomials calculator free?
Multiplying polynomials is the one topic in algebra where being systematic beats being quick. Count the products, take them in an order you trust, keep the collecting separate from the multiplying, and check the degree at the end. Use the calculator above to see the grid a few times and the habit forms on its own.