Late September 2026: numbers and sets are wrapping up — powers, GCD and LCM, fractions, ratios, sets and logic — and algebra starts in November. Special products arrive between the end of November and December, with the test between December and January: eight to twelve weeks from now. This applies at the classical high school (liceo classico), where maths has three hours a week and few guided exercises are done in class, but it applies in exactly the same way at the science, languages and technical schools: the chapter doesn’t change with the track.
On monomials and polynomials in general I have already written a separate guide, aimed at those who come from a vocational school with gaps on signs: monomials and polynomials in Year 1, which touches special products only in passing. Here I do the opposite: I take the operations between monomials and polynomials as given, and go in depth only on special products — the ones the classical high school covers with fewer exercises in class and more independent reading of the book.
Why they are called “special”: an identity, not a trick
A special product (prodotto notevole) is a particular case of multiplication between polynomials that, instead of requiring the full term-by-term calculation, can be written directly in an already expanded form — an identity that is always true, not an approximation. The reason it is worth learning them by heart is not laziness: it is that they come back all the time, in geometry, in physics, and later in factorisation, where they are used to read an expression “backwards”.
The square of a binomial, and why it is not
The most widespread mistake of the whole first year is stopping at the first and last terms, forgetting the double product . With , you see at once how much that missing piece weighs:
Forty-nine, not twenty-five: it is not a rounding, it is a different number. There is also a geometric reading, useful so as not to treat the formula as an arbitrary rule: is the area of a square with side . That square breaks down into a square with side (area ), one with side (area ), and two rectangles with sides and (area each, so in all) — exactly the term that is missing for anyone who stops at .
When one of the two terms is negative, as in , the double product changes sign but stays negative regardless of which of the two terms was negative to start with: both and give , because squaring loses the sign of the starting binomial.
Spotting the “disguised” square of a binomial
The place where most marks are lost in a test is not applying the formula to written exactly like that, but recognising it when the terms are not and but more complex expressions: coefficients, signs, several letters together.
Here and : the typical mistake is to square only the letter and forget the coefficient, writing instead of (the coefficient must be squared just like the , becoming ).
With several letters in the same monomial it works identically:
Sum times difference:
You recognise it by a precise signal: the same two terms, one with a and one with a in the middle. If even just one of the two terms changes between the two brackets, it is no longer this special product and must be worked out in full like a product of any two binomials.
The opposite case — mistaken for because “after all there’s an and a ” — is the twin of the square of a binomial written without the double product: here the double product cancels by itself (), and whoever doesn’t notice risks adding it anyway out of habit, getting a wrong result with a term in that isn’t there.
The square of a trinomial: three terms, three double products
When the binomial becomes a trinomial, the square widens: one double product is no longer enough, you need three, one for each pair of terms.
A way not to forget any: every term squared, plus the double product of every possible pair taken from the three terms (with its own sign). With three terms the possible pairs are three — , , — and that is why exactly three double products appear, not two or four.
Here , , : the term carries the minus sign, and and carry theirs accordingly. Always writing the three terms with their own sign before adding, without working out the double products in your head, is the most reliable way not to lose one along the way.
The cube of a binomial
Unlike the square of a binomial, in the cube the signs are not lost when one of the two terms is negative: in the version with the minus, the signs alternate (plus, minus, plus, minus), because odd powers of stay negative. This is why it is worth remembering the version with the minus as an identity in its own right, not as “the same as before with a generic sign in front of ”.
Tartaglia’s triangle: where those coefficients come from
The coefficients of the cube of a binomial are not arbitrary: they are the fourth row of Tartaglia’s triangle (known in English as Pascal’s triangle), the same structure that generates the coefficients of the expansion of for every . Each row begins and ends with , and each inner number is the sum of the two numbers above it in the previous row:
The row is that of the square of a binomial (), the row that of the cube. If one day you need the fourth power of a binomial, the next row () gives it to you without having to multiply four times in full — although in Year 1 you usually stop at the cube.
Reading the formula backwards: a preview of factorisation
All the special products seen so far also work read backwards: from an already expanded polynomial you can go back to the compact form, if the polynomial has exactly the right structure. is not “any old polynomial that doesn’t factorise”: it is a difference of squares, , so it is rewritten as . In the same way, is not a random trinomial: it is , that is exactly the expansion of — the so-called perfect-square trinomial, recognisable because the first and third terms are perfect squares and the second is their double product.
This backwards reading is the heart of the chapter that follows special products, factorisation: I go into it more deeply, with the full decision tree for recognising every case, in factorisation and algebraic fractions in Year 2 — it is aimed at Year 2; the method for starting in Year 1, with worked exercises on the most common cases, is in factorising polynomials: method and worked exercises, and the underlying logic (recognise the structure before factorising) is the same one that already applies here, in reverse.
Typical mistakes
| Mistake | How you notice it |
|---|---|
| , double product forgotten | Try it with two real numbers, for example and : if the two results differ, is missing |
| Coefficient not squared along with the letter, in or similar | The coefficient is a factor inside the bracket like the others: if you square (or cube) the letters, do the same to it |
| mistaken for , or vice versa | Look at the sign between the two brackets: the same in both is a square, different is a sum times difference — and in that case the double product cancels, it must not be added |
| A double product forgotten in the square of a trinomial | Count the possible pairs among the three terms: there are always three (, , ), write them one by one before adding |
| Signs lost in the cube of a binomial with the minus | Remember they alternate (plus, minus, plus, minus): it is not “the square with a generic minus sign” |
Three worked exercises
1. Disguised square of a binomial. Expand .
I recognise , :
The coefficient must be squared like the letter that goes with it, becoming ; the same for the in , which becomes .
2. Cube of a binomial. Expand .
I use with , :
3. An expression with several special products together. Simplify:
I work out the three special products separately, then add:
I substitute, paying attention to the minus sign in front of the second bracket, which must be distributed over both its terms:
I group the like terms:
The minus sign in front of is where most is lost: it changes the sign of both terms inside the bracket, turning into , not leaving it as .
You can practise more exercises like these in the maths test — monomials and special products: the correct answer is free, and the guided step-by-step solution unlocks with your first lesson.
Before the test
If this is the first maths test of the year or close to it, I have written a more general guide on how to prepare for the first Year 1 test, with the full picture of what comes up and a study plan for the rest of the syllabus. The full programme for the year is at /materie/matematica/.
If you want a hand before the December-January test, the first introductory call is free and lasts half an hour.
