Free · Year 1 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 1: factorising.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Taking out a common factor, in full and by grouping
Difference of squares, square and cube of a binomial
Sum and difference of cubes
Factorising trinomials, including a leading coefficient other than 1
Ruffini’s rule and the remainder theorem
HCF and LCM of polynomials
Algebraic fractions: domain restrictions, simplifying, sum and product

Two worked examples from the test

Question 1 · Scomposizione in fattori e frazioni algebriche · Scomposizione in fattori

Scomponi raccogliendo a fattor comune totale:

$12 a^{4} b - 18 a^{3} b^{2}$

A $6 a^{3} b (2 a - 3 b)$ correct answer
B $2 a^{3} b (6 a - 9 b)$
C $6 a^{3} b (2 a - 3 b^{2})$
D $6 a^{4} b (2 - 3 b)$

Solution

Il fattore comune numerico è il MCD dei coefficienti: $MCD (12, 18) = 6$ .
Per ogni lettera si prende l'esponente minore: $a^{3}$ e $b^{1}$ .
Il fattore comune totale è quindi $6 a^{3} b$ .
Dividi ogni termine: $12 a^{4} b : 6 a^{3} b = 2 a$ e $18 a^{3} b^{2} : 6 a^{3} b = 3 b$ .
Risultato: $6 a^{3} b (2 a - 3 b)$ .

Question 2 · Scomposizione in fattori e frazioni algebriche · Scomposizione in fattori

Scomponi con il raccoglimento a fattor parziale:

$3 x^{3} + 3 x^{2} - 5 x - 5$

A $(x + 1) (3 x^{2} + 5)$
B $(x - 1) (3 x^{2} - 5)$
C $(x + 1) (3 x^{2} - 5)$ correct answer
D $(x + 1) (3 x - 5)$

Solution

Raggruppa i termini a due a due: $(3 x^{3} + 3 x^{2}) + (- 5 x - 5)$ .
Dal primo gruppo raccogli $3 x^{2}$ : $3 x^{2} (x + 1)$ .
Dal secondo gruppo raccogli $- 5$ (non $+ 5$ , altrimenti la parentesi non coincide): $- 5 (x + 1)$ .
Ora $(x + 1)$ è comune: $3 x^{2} (x + 1) - 5 (x + 1) = (x + 1) (3 x^{2} - 5)$ .

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 4 pages · 80 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on factorising looks like

Five or six polynomials to factorise, of increasing difficulty, then an algebraic fraction to simplify and a sum of algebraic fractions with the domain restrictions. Factorising is the bridge: if it does not hold, the algebraic fractions never get started.

The mistake that costs the grade

Stopping at the first step. Taking out a common factor almost never finishes the exercise: inside the brackets there is nearly always a difference of squares or a trinomial that can still be factorised, and the teacher takes marks off for incomplete factorising.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read how to study maths without stress — or we look at your results together: the diagnostic is free.

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