Algebra: monomials and polynomials

Mid-September 2026: classes started a few days ago, and in Year 1 of an Italian vocational school (istituto professionale) the review of number sets and fractions is about to give way to monomials, between the end of September and the start of October. The first test on algebra arrives in five to eight weeks, so between the end of October and mid-November — and in many schools it is the first test of the year in which maths stops being “just numbers” and turns into letters.

Why a different approach makes sense here

Algebraic calculation in maths, together with sentence and clause analysis in Italian, is one of the two blocks that generate the first failed subject in Year 1. It is not an isolated case: in the first two years of vocational schools the risk of leaving school is structurally higher than average, and the first months of maths are one of the moments when it is decided whether that risk stays theoretical or becomes real. That said, there is no reason to live it with anxiety: it can be recovered, and recovered in a few weeks if you work on the right block.

Many students arrive here with real gaps from middle school — in fractions, in signs, in the order of operations — and algebra brings them all out at once, because every earlier mistake repeats identically, only with a letter in place of the number. It is not the fault of whoever arrives with those gaps: it is simply the point at which, if you have never worked on signs methodically, it shows.

The key to not treating letters as an abstract game is to start from something concrete. A letter, in these exercises, is almost always a quantity that changes: the cost of $n$ place settings in a restaurant, the length of $n$ pieces cut from the same pipe, the discounted price of a service. First you understand what the letter stands for in the problem, then you apply the rule — not the other way round.

If the first term goes wrong, the first two years of vocational schools have a tool designed for exactly this: the PFI, the Individual Training Plan (Progetto Formativo Individuale), with hours of personalisation set aside for those who need to catch up. It is worth knowing about, without making a drama of it: it is the tool the school itself provides when a failed subject opens early.

What a monomial is

A monomial is an expression with a numerical part (the coefficient) and a letter part made only of letters raised to powers, multiplied together — never added, never divided by a letter. $5 a^{2} b$ is a monomial: coefficient $5$ , letter part $a^{2} b$ . The degree is the sum of the exponents of the letters: here, $2 + 1 = 3$ .

Two monomials are like terms (simili) when they have the same letter part, that is the same letters with the same exponents. $3 a^{2} b$ and $- 7 a^{2} b$ are like; $3 a^{2} b$ and $3 a b^{2}$ are not, even though they look “almost the same”.

Algebraic sum: you only add like monomials

This is mistake no. 1 across the whole topic: adding different letter parts as if they were the same thing.

3 a + 2 b \neq = 5 ab

$3 a$ and $2 b$ are not like terms — one letter is $a$ , the other is $b$ — so they stay written separately: $3 a + 2 b$ is already the final result, it can’t be squeezed into a single term. You only add (or subtract) the coefficients of monomials with exactly the same letter part:

3 a^{2} + 5 a^{2} - 2 a^{2} = 6 a^{2}

Product, quotient and power of a monomial

In a product of monomials you multiply the coefficients and add the exponents of equal letters: $2 a^{2} \cdot 3 a^{3} = 6 a^{5}$ . In a quotient you divide the coefficients and subtract the exponents: $6 a^{5} : 2 a^{2} = 3 a^{3}$ .

In a power of a monomial, the outer exponent applies to the whole monomial — coefficient included, not just the letters. It is the second most widespread mistake:

(2 a^{2})^{3} = 2^{3} \cdot a^{2 \cdot 3} = 8 a^{6} not 2 a^{6}

Whoever writes $2 a^{6}$ has cubed only the letter and left the coefficient unchanged — exactly like forgetting that the $2$ is also “inside” the bracket.

GCD and LCM of monomials are worked out as for numbers, but taking the letters into account as well: the GCD takes, for each common letter, the lowest exponent and the greatest common numerical factor of the coefficients; the LCM takes the highest exponent and the least common multiple of the coefficients. Between $12 a^{2} b$ and $18 a b^{3}$ , the GCD is $6 ab$ and the LCM is $36 a^{2} b^{3}$ .

Polynomials: sum, product, and the two steps that combine

A polynomial is a sum of monomials that are not like terms. Its degree is that of the highest-degree monomial it contains.

To multiply a monomial by a polynomial you apply the distributive property, multiplying the monomial by each term:

2 a \cdot (3 a^{2} - 4 a + 1) = 6 a^{3} - 8 a^{2} + 2 a

To multiply two binomials, you multiply each term of the first by each term of the second, then add the like terms:

(x + 2) (x - 5) = x^{2} - 5 x + 2 x - 10 = x^{2} - 3 x - 10

Special products: what looks like a shorter operation, isn’t

The square of a binomial is the shortcut most badly used in the whole of Year 1. $(a + b)^{2}$ is not $a^{2} + b^{2}$ : a piece is missing, the double product.

(a + b)^{2} = a^{2} + 2 ab + b^{2}

With $a = 3$ and $b = 4$ you see at once how much that missing piece matters:

(3 + 4)^{2} = 7^{2} = 49 3^{2} + 4^{2} = 9 + 16 = 25

Forty-nine against twenty-five: it is not a rounding, it is a different number. Geometrically, $(a + b)^{2}$ is the area of a square with side $a + b$ : it breaks down into a square with side $a$ (area $a^{2}$ ), one with side $b$ (area $b^{2}$ ), and two rectangles with sides $a$ and $b$ (area $ab$ each) — hence the $2 ab$ that is always missing for anyone who stops at $a^{2} + b^{2}$ .

The other special product of Year 1 is the difference of squares:

(a + b) (a - b) = a^{2} - b^{2}

Three exercises worked step by step

Exercise 1 — like monomials with negative signs. Simplify:

5 a^{2} - 3 a + 2 a^{2} - 7 a - a^{2}

First group the terms with $a^{2}$ , then those with $a$ :

(5 a^{2} + 2 a^{2} - a^{2}) + (- 3 a - 7 a) = 6 a^{2} - 10 a

The result doesn’t simplify any further: $6 a^{2}$ and $- 10 a$ are not like terms, so they remain two separate terms.

Exercise 2 — power of a monomial. Simplify $(2 a^{2} b)^{3}$ .

The exponent $3$ applies to every factor inside the bracket: to the coefficient $2$ , and to each letter with its own exponent multiplied by $3$ .

(2 a^{2} b)^{3} = 2^{3} \cdot a^{2 \cdot 3} \cdot b^{1 \cdot 3} = 8 a^{6} b^{3}

Exercise 3 — square of a binomial, applied case. The bill for a table at a restaurant is given by $(x + 4)^{2}$ , where $x$ is the price in euros of the basic dish and $4$ the cover charge per person. If $x = 6$ , how much does the table pay — and how far off would someone be who calculates $x^{2} + 16$ instead of the square of the binomial?

The correct calculation:

(x + 4)^{2} = x^{2} + 8 x + 16 ⟹ (6 + 4)^{2} = 1 0^{2} = 100

Whoever uses the wrong shortcut calculates instead:

x^{2} + 16 = 36 + 16 = 52

One hundred euros against fifty-two: almost half. It is the same mistake as in the example with $a = 3$ , $b = 4$ seen above, applied to a number that here stands for real money — the reason it is worth always checking with a numerical case before trusting the formula from memory.

You can practise more algebra exercises 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. An honest warning: that practice test is calibrated on the high school (liceo) syllabus, while at vocational schools the algebra itself is the same but the applied problems are different — feel free to skip the questions that seem off-syllabus and concentrate on the pure calculation exercises.

Typical mistakes, and how to catch them yourself

Typical mistake	How you notice it
Adding unlike monomials ( $3 a + 2 b \to 5 ab$ )	Check whether the two letter parts are really identical: different letters, or the same letter with different exponents, never add up
Forgetting the sign in front of a bracket	If there is a minus in front, every term inside the bracket changes sign when you remove it: rewrite them one by one, not in your head
The exponent that doesn’t touch the coefficient ( $(2 a^{2})^{3} = 2 a^{6}$ )	The coefficient is a factor like the others inside the bracket: if you cube the letters, cube it too
$(a + b)^{2}$ expanded as $a^{2} + b^{2}$	Try it with two real numbers, like $3$ and $4$ : if the two results differ, the double product $2 ab$ is missing
Wrong signs in the product of two binomials	Write all four products before adding, without skipping any in your head: that is where a sign gets lost

A three-week catch-up plan, 20-25 minutes a day

You don’t need to turn your afternoons upside down to catch up: three weeks with a little time every day is enough, as long as it is targeted.

Week 1 — monomials. Every day, ten minutes of theory (definition, degree, like terms) and the rest in exercises: sum of like monomials, then product and power. The goal of the week is to recognise at a glance whether two monomials are like before even trying to add them.
Week 2 — polynomials. Product of a monomial by a polynomial, then product of two binomials, every day with at least five different expressions. Here you see straight away whether Week 1 left any holes: if the product of two binomials goes wrong, often the problem is not the product, it is still the sum of like monomials at the end.
Week 3 — special products and applied problems. Square of a binomial and difference of squares, with at least one exercise a day tied to a concrete context (a cost, a measurement, a discount), and in the last days a timed mock test, under the same conditions as the real test.

How we work in lessons

In lessons I don’t redo the textbook theory: I start with a five-minute mini-test to find out whether the shaky block is still in the numbers (signs, fractions) or already in algebra, and I work only there, with immediate correction at every step. If we haven’t met yet, there is a free 30-minute introductory call to work out where to start, no strings attached.

For the following year in the same track I wrote equations, inequalities and systems in Year 2 at vocational school. For this year’s parallel subject there is law and economics in Year 1: sources of law and legal capacity. And if you want to see other algebraic shortcuts that look like real rules and aren’t, there is five maths mistakes that look right. The full syllabus for the year is on the maths page. For the special-product formulas with more worked exercises there is special products: formulas and worked exercises.

If you would like a hand before the end-of-October test, write to me: three weeks of targeted work now are worth more than a failed subject to recover in January.

Monomials and polynomials in Year 1 of high school