Factorising and the line: the first Year 2 test

School started again a few days ago. The first Year 2 maths test, at the Italian technical institutes, almost always arrives between mid-October and early November — four to six weeks from today. The syllabus is almost always the same everywhere: a review of factorising polynomials, then the Cartesian plane and the straight line. These are the topics where the teacher starts again from scratch, but also the ones where the Year 1 failed subject was formed: almost always in January–February, precisely on polynomials and factorising. If that chapter stayed fragile in Year 1, it is here that you see it straight away.

This post covers the two blocks with the same method I use in lessons: first the mistakes that lose marks — with numbers, not in the abstract — then the step-by-step procedure, then three exercises worked in full.

What the first test really asks

Under “factorising”, a Year 2 test at a technical institute generally asks for: taking out a common factor, factorising by grouping, difference of squares, square of a binomial (sometimes cube of a binomial), sum or difference of cubes, the special trinomial $x^2+sx+p$, and at least one exercise where these steps are used to simplify an algebraic fraction — with the conditions of existence written down, not just thought.

Under “line” you are expected to: write the equation of the line through two points, calculate the slope, recognise parallel and perpendicular lines, and often a mixed exercise asking for the distance between two points or the midpoint of a segment.

They are two different chapters but the test almost always puts them together, because the second part of the syllabus (the Cartesian plane) is built on the first (algebra): a sign error in factorising comes back identically when you calculate a slope with a fraction.

The mistakes that cost marks

1. Factorising without taking out the common factor first. Take $2x^3-8x$: the most common mistake is to start straight away with the difference of squares on $x^3-4x$, skipping the common factor 2. The right method is always the same order: first take out the common factor, always — even when it seems there is nothing to take out.

Anyone who skips the first step often also stops at the second, handing in $2x(x^2-4)$ as if it were already fully factorised: it isn’t, because $x^2-4$ is still a difference of squares.

2. Wrong signs when factorising by grouping. In $ax-ay-bx+by$ the problem is not taking out a factor, it is the sign with which you take out the second group:

The trap is writing $-b(x-y)$ as $-b(x+y)$ or forgetting that the minus sign in front of $bx$ has to be taken out together with the factor, not left inside the bracket.

3. Confusing $a^2-b^2$ with $a^2+b^2$. $a^2-b^2=(a-b)(a+b)$ can always be factorised. $a^2+b^2$ cannot be factorised over $\mathbb{R}$: there is no pair of real factors whose product gives a sum of two positive squares. Seeing a plus instead of a minus and factorising anyway is one of the mistakes a marker notices first.

4. The sign of the constant term in the special trinomial. To factorise $x^2+sx+p$ in the form $(x+a)(x+b)$ you need two numbers $a$ and $b$ such that $a\cdot b=p$ and $a+b=s$. The typical mistake is to look for two numbers that add up to $s$ and then check the product by guesswork, instead of starting from the product: if $p$ is negative, $a$ and $b$ have opposite signs; if $p$ is positive, they have the same sign as $s$.

5. Forgotten conditions of existence. On an algebraic fraction like $\frac{x^2-4}{x^2-2x}$, simplifying without first writing the conditions of existence (C.E.; here $x\ne 0$ and $x\ne 2$, because the denominator vanishes there) is a mistake that many teachers consider as serious as a wrong calculation, because it changes the domain of the function, not just the look of the expression.

The method for factorising, step by step

1. Always take out the total common factor, if there is one — numerical and literal.
2. Count the terms that are left: two terms → try difference of squares or sum/difference of cubes; three terms → try square of a binomial or the special trinomial; four terms → try grouping in pairs.
3. After each factorisation, look again at every factor you got: can it still be factorised? The classic case is $x^4-16=(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)$, where $x^2+4$ is not touched any further but $x^2-4$ is.
4. If an algebraic fraction appears, factorise numerator and denominator, write the C.E. from the factorised denominator, then cancel the common factors.

The method for the line, step by step

Given two points $A(x_1,y_1)$ and $B(x_2,y_2)$, the slope of the line joining them is

Once you have found $m$, the equation of the line is written in explicit form $y=mx+q$ by substituting the coordinates of one of the two points to find $q$, or it stays in implicit form $ax+by+c=0$ when it is convenient not to isolate $y$ — for example precisely in the case of vertical lines $x=k$, which have no explicit form: there is no $m$ for a line parallel to the $y$ axis.

Two lines are parallel if they have the same slope, $m_1=m_2$. They are perpendicular if the product of the slopes is $-1$:

The midpoint of segment $AB$ is the average of the coordinates, $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$, and the distance between $A$ and $B$ is

The most frequent mistakes here: reading the sign of $m$ backwards from a graph (a line that goes down from left to right has a negative $m$, not positive), getting the sign wrong when going from implicit to explicit form when the coefficient of $y$ is negative, and confusing “the point lies on the line” (you substitute the coordinates into the equation and a true equality must come out) with “it is a solution of a system” (which concerns the intersection of two lines, not whether a point belongs to a single line).

Three worked exercises

Exercise 1 — mixed factorising. Factorise $3x^3-12x$.

First step, take out the common factor: the common factor is $3x$.

The factor $x^2-4$ is a difference of squares, so it factorises further:

Stop here: none of the three factors can be factorised further.

Exercise 2 — algebraic fraction with C.E. Simplify $\frac{x^2-x-6}{x^2-9}$.

The numerator is a special trinomial: I look for two numbers whose product is $-6$ and whose sum is $-1$, that is $-3$ and $2$.

The denominator is a difference of squares:

Before simplifying, the conditions of existence: the original denominator vanishes for $x=3$ and $x=-3$, so C.E.: $x\ne 3\wedge x\ne -3$. Now we cancel the common factor $(x-3)$:

The condition $x\ne 3$ stays written even after the simplification, because otherwise the final expression would seem defined there too, while the starting one was not.

It is worth doing extra practice on this type of exercise before the test: in /simulazioni/verifiche-liceo/ there are tests organised by topic, not mixed by year — precisely so you can train on factorising and algebraic fractions separately from the line.

Exercise 3 — line through two points and perpendicular. Given $A(1,2)$ and $B(3,8)$, write the equation of the line $r$ through $A$ and $B$, then the equation of the line $s$ perpendicular to $r$ passing through $C(0,5)$.

Slope of $r$:

Equation of $r$ in explicit form, using the point $A$:

Quick check with $B$: $3\cdot 3-1=8$, correct.

For the perpendicular $s$ we need $m_s$ such that $m_r\cdot m_s=-1$, so $m_s=-\frac{1}{3}$. Passing through $C(0,5)$:

$C$ has $x=0$, so $q=5$ can be read directly: it is the $y$-intercept.

Why you need it straight away, not just for the grade

On the economics track, the line is the first model you meet in business studies (Economia aziendale): total cost and revenue as functions of the quantities produced or sold are, in the simplest cases, exactly functions of the type $y=mx+q$, with $m$ representing a unit cost or price and $q$ a fixed cost. On the technology track, the same tools — slope, reading a graph, the equation of a line — come back identically in the distance–time and velocity–time graphs of physics, where the slope of the graph is a velocity or an acceleration. It is not an isolated chapter of maths: it is the language that the specialist subjects will use from here on.

How we work in lessons

With students who arrive with the Year 1 failed subject still uncovered or with shaky foundations on these two chapters, the work starts from where understanding stopped — often precisely the sign in grouping or reading $m$ on a graph — and builds from there towards test-type exercises. You will find the full Year 2 syllabus, with all the chapters of the year, in Year 2 of high school: maths syllabus and formula sheet; after the line come linear systems, which use it to find the intersection of two lines, and later in the year radicals. If you want a cross-cutting review of the mistakes that come back chapter after chapter, there is also Five maths mistakes that look right. When reading $m$ and $q$ is solid, the next step is pencils of lines, with worked exercises in pencils of lines: proper and improper.

Maths lessons are online and one-to-one. We start with a free 30-minute introductory call to see where you stand; the rates (single lesson, 10-hour package) are on the pricing page. Payment is always offline — bank transfer, Satispay, PayPal, card, Apple Pay or Revolut.

To practise on your own before the test, the tests organised by topic are in /simulazioni/verifiche-liceo/: the correct answer is free, the guided solution is unlocked with your first lesson. If you would rather talk about it first, we are on /contatti/.