Algebra · Paper 1
Introduction
Higher Level · Tap NEXT to begin
Section 1 of 5
The Savings Story
Forget formulas for a minute. A boy has €3 today and saves another €2 every day. We’ll build a table, plot the points, spot the pattern, write the equation.
That single page of work introduces four pieces of vocabulary you’ll meet in every chapter that follows.
Three definitions before we start
Independent variable
The one you choose — usually time, or whatever sits on the horizontal axis. Call it $x$.
Dependent variable
The one that responds — usually what you’re measuring. Call it $y$. Its value is a function of $x$, written $y = f(x)$.
Initial value
The value when $x = 0$. Also called the $y$-intercept, the origin, or just the starting amount. In the savings story, that’s the €3 already in the box.
Worked example — from a story to a function
A boy has €3 saved and adds €2 every day.
(i) Make a table for the first five days.
(ii) Plot the points on a Cartesian plane.
(iii) Write an equation for the total savings $y$ after $x$ days.
(i) Build the table — watch the gap
Time is what we choose: independent, call it $x$. Money is what responds: dependent, call it $y$. Start at day 0.
| Time (days) $x$ | Money (€) $y$ |
|---|---|
| 0 | 3 |
| 1 | 5 +2 |
| 2 | 7 +2 |
| 3 | 9 +2 |
| 4 | 11 +2 |
| $x$ | $2x + 3$ |
The gap
Each row’s $y$ is two more than the one before. That constant gap — $d = 2$ in arithmetic-sequence language — will turn out to be the slope of our line. Same number, three different names. Watch it appear again in the next section.
(ii) Plot the points — they sit on a straight line
Each row of the table is a coordinate pair $(x, y)$. Plot $(0, 3)$, $(1, 5)$, $(2, 7)$, $(3, 9)$, $(4, 11)$. Join them — the points line up.
Every point sits on the same straight line. The rise (2) over the run (1) is the slope.
Read the picture
The line cuts the $y$-axis at 3 — that’s the initial value, the $t = 0$ moment. For every one step to the right (one day), the line goes up two (two euro). That’s the gap from the table, drawn as a triangle.
(iii) Write the equation — start with 3, add 2 per day
In words: total savings = starting amount + (gap × number of days).
$y = 3 + 2x$
Tidied:
$y = 2x + 3$
Using function notation:
$y = f(x) = 2x + 3$
What just happened
Three pieces of vocabulary fell out of one story: the gap in the table (2) became the slope of the line, and the initial value (3) became the $y$-intercept. Same numbers, different language, same picture. That repetition is the whole game.
Done. $y = f(x) = 2x + 3$. Memorise the shape: output = (gap) × input + (start). Every linear function in this chapter is that sentence in disguise.
Section 2 of 5
Slope — one idea, six names
The €2 in the savings story has more aliases than a spy in a film. Same number every time — 2 — just dressed differently for each chapter. Learn the names now and the rest of the course stops feeling like six separate topics.
The six names of slope
Every line you’ll meet in this course has one number that controls how steep it is. Depending on which chapter you’re reading, that number goes by a different name.
$m$
Just the letter we use for slope, the way $c$ is the letter we use for a constant. $y = mx + c$.
Rise over run
Pick any two points on the line, divide vertical change by horizontal change. $m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}$.
Rate of change
In plain English, how fast $y$ changes per unit of $x$. €2 per day. 60 km per hour. 0.05 m per second.
$\dfrac{dy}{dx}$
The calculus name for exactly the same thing, used in differentiation. We’ll meet it later — same number, fancier hat.
Gap, $d$
In an arithmetic sequence, the constant difference between consecutive terms. In the savings table: $d = 2$.
All six names point at the same €2. When this chapter says slope, the sequences chapter says common difference, and the calculus chapter says derivative. Same number, every time.
Worked example — three names, same number
For the savings function $y = 2x + 3$:
(i) compute the slope using rise over run from $(0, 3)$ to $(3, 9)$.
(ii) state the gap $d$ between consecutive table entries.
(iii) describe the rate of change in everyday words.
(i) Rise over run from two points
Pick the two points and apply the formula.
$m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} = \dfrac{9 - 3}{3 - 0} = \dfrac{6}{3} = 2$
(ii) The gap between consecutive table rows
From the savings table: $5 - 3 = 2$, $7 - 5 = 2$, $9 - 7 = 2$. Constant.
$d = 2$
Same answer
The gap $d$ in the table is the slope $m$ on the graph. Not “related to” — equal to. They’re the same number sitting in two pieces of vocabulary.
(iii) In plain English, with units
The boy adds €2 per day. That’s the rate of change of money with respect to time.
Rate of change = €2 per day
$m = d =$ rate of change $= \mathbf{2}$ (€ per day)
Done. Three calculations, three names, one number. When the next chapter calls it a common difference, or the one after calls it $\dfrac{dy}{dx}$, you’ll know it’s the same 2.
Section 3 of 5
Anatomy of a Polynomial
The savings function $y = 2x + 3$ has two terms. Most algebra in this book has more — up to four or five. Every term has a name for each piece, and the whole expression has a name based on how many terms it has and the highest power.
Learn these labels once and every question in every chapter reads like a sentence.
Expression. Function. Polynomial.
An object like $5x^{4} + 6x^{3} + 2x^{2} + 7x + 9$ — a sum of terms each of the form (number) × (variable to a whole-number power) — gets called by three names depending on context:
Expression
The general name when you’re just looking at it or simplifying it.
Function
The name when you’re plugging in values of $x$ to get values of $y$. We write $f(x) = 5x^{4} + 6x^{3} + 2x^{2} + 7x + 9$.
Polynomial
The formal name. Means “many terms”, from the Greek poly- + -nomial.
Inside a single term — for $5x^{4}$, three labels
Coefficient
The number in front. Here, $5$.
Variable
The letter. Here, $x$.
Power = Index = Degree
The little number up top. Here, $4$. Three names for one number. Learn all three because exam papers swap between them at will.
And the lone number at the end — the one with no $x$ at all — is called the constant. In $5x^{4} + 6x^{3} + 2x^{2} + 7x + 9$, the constant is $9$.
Naming the whole polynomial — count the terms, find the highest power
By number of terms:
Monomial
1 term. Example: $7x^{3}$.
Binomial
2 terms. Example: $2x + 3$.
Trinomial
3 terms. Example: $3x^{2} + 7x + 9$.
By highest power (the degree of the polynomial):
Linear — degree 1
$2x + 3$. Graph: a straight line.
Quadratic — degree 2
$3x^{2} + 7x + 9$. Graph: a parabola.
Cubic — degree 3
$x^{3} - 4x + 1$. Graph: an S-shape.
Quartic — degree 4
Example: $5x^{4} + 6x^{3} + 2x^{2} + 7x + 9$.
A polynomial almost always wears both labels — “quadratic trinomial” means three terms with highest power 2.
Worked example — putting the vocabulary on
For $5x^{4} + 6x^{3} + 2x^{2} + 7x + 9$:
(i) how many terms?
(ii) state the coefficient and degree of each term, and identify the constant.
(iii) name the polynomial by both its term-count and its degree.
(i) Count the terms — separated by + or −
$5x^{4}$ | $6x^{3}$ | $2x^{2}$ | $7x$ | $9$
5 terms.
“Parts” and “terms” mean the same thing
Some books say “this polynomial has 5 parts”, others say “5 terms”. Identical.
(ii) Pull each term apart
| Term | Coefficient | Variable | Degree |
|---|---|---|---|
| $5x^{4}$ | 5 | $x$ | 4 |
| $6x^{3}$ | 6 | $x$ | 3 |
| $2x^{2}$ | 2 | $x$ | 2 |
| $7x$ | 7 | $x$ | 1 |
| $9$ | 9 | — | 0 |
The last row — the term with no $x$ — is the constant. It’s still a term; it just has degree 0 (because $x^{0} = 1$ for any $x$, so $9 = 9 \cdot x^{0}$).
Power, index, degree
The little superscript number has three names. Same number every time. In exam language: “the index of $5x^{4}$ is 4”, “the degree of $5x^{4}$ is 4”, “the power of $5x^{4}$ is 4”. Pick one and stick with it — recognise the other two.
(iii) Name the whole thing
By term-count: 5 terms — no special Greek name; we’d just say “a polynomial in five terms”.
By degree: highest power is 4, so it’s a quartic.
A quartic polynomial in 5 terms.
Done. Coefficient · variable · power. Constant at the end. Count terms · find the highest power. Every algebra question that mentions a polynomial will lean on at least one of those labels.
Section 4 of 5
Where $x$ Lives — the four number systems
Every time we wrote $x$ in this lesson, we assumed $x$ was a number. But which kind? The savings story uses whole days $(0, 1, 2, \ldots)$, but most algebra questions allow fractions, negatives, even irrationals like $\sqrt{2}$.
There’s a Russian-doll structure of four named sets — learn the symbols now, they’ll appear on every paper.
Natural numbers — $\mathbb{N}$
$\mathbb{N} = \{1, 2, 3, 4, \ldots\}$
Whole, positive. Counting numbers. $x \in \mathbb{N}$ = whole, positive.
Integers — $\mathbb{Z}$
$\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$
Whole numbers, positive and negative, plus zero. $x \in \mathbb{Z}$ = whole, positive or negative.
Rational numbers — $\mathbb{Q}$
$\mathbb{Q} = \left\{ \dfrac{p}{q} : p, q \in \mathbb{Z}, \; q \ne 0 \right\}$
Anything that can be written as one integer over another. Fractions. $x \in \mathbb{Q}$ = fractions $(p/q)$.
Real numbers — $\mathbb{R}$
$\mathbb{R}$
Every number that exists on the number line. Includes irrationals like $\sqrt{2}$, $\pi$, $e$. $x \in \mathbb{R}$ = any number that exists.
Each set sits inside the next
Every natural number is also an integer. Every integer is also a rational (write it as $\dfrac{n}{1}$). Every rational is also a real. So:
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
Each circle is contained in the next. $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. The shell $\mathbb{R} \setminus \mathbb{Q}$ is the irrationals — numbers like $\sqrt{2}$, $\pi$, $e$.
The numbers in $\mathbb{R}$ that are not in $\mathbb{Q}$ — written $\mathbb{R} \setminus \mathbb{Q}$ — are the irrationals. They can’t be written as $p/q$. Examples: $\sqrt{2}$, $\sqrt{3}$, $\pi$, $e$.
Worked example — classify each number into the smallest set it belongs to
For each, name the smallest of $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ it belongs to:
(i) $7$, (ii) $-4$, (iii) $\dfrac{2}{3}$, (iv) $\sqrt{2}$, (v) $0$, (vi) $\sqrt{9}$.
(i) $7$
A counting number. Smallest set: $\mathbb{N}$.
(Of course 7 is also in $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ — but $\mathbb{N}$ is the smallest.)
(ii) $-4$
Whole, but negative. Not in $\mathbb{N}$. Smallest set: $\mathbb{Z}$.
(iii) $\dfrac{2}{3}$
A fraction — integer over integer. Not whole, so not in $\mathbb{Z}$. Smallest set: $\mathbb{Q}$.
(iv) $\sqrt{2}$
$\sqrt{2} \approx 1.41421356\ldots$ — the decimal never repeats and never ends. Cannot be written as $p/q$. Irrational.
Smallest set: $\mathbb{R}$.
(v) $0$
Whole, but not positive. Many books exclude $0$ from $\mathbb{N}$. Smallest set: $\mathbb{Z}$.
Convention
In the Irish syllabus, $\mathbb{N} = \{1, 2, 3, \ldots\}$ — does not include $0$. Some other countries do include it. Match whatever your textbook says; for State Exams, $0$ is in $\mathbb{Z}$, not $\mathbb{N}$.
(vi) $\sqrt{9}$ — simplify first
$\sqrt{9} = 3$. The surd looks irrational, but it isn’t — it’s just $3$ in disguise.
Smallest set: $\mathbb{N}$.
Always simplify before classifying
$\sqrt{9}$ looks like an irrational, but it’s 3. $\sqrt{8}$ looks like an irrational, and it is one (it simplifies to $2\sqrt{2}$, still irrational). Don’t be tricked by the surd sign.
(i) $\mathbb{N}$ (ii) $\mathbb{Z}$ (iii) $\mathbb{Q}$ (iv) $\mathbb{R}$ (v) $\mathbb{Z}$ (vi) $\mathbb{N}$
Done. Russian dolls. Work outwards from $\mathbb{N}$ until the number fits. If nothing fits until $\mathbb{R}$, it’s irrational. Simplify any surds before you classify — that’s the only trap.
Section 5 of 5
Four ideas. Every chapter ahead uses them.
The savings story gave us function notation: $y = f(x)$, an independent variable and a dependent one.
Slope goes by six names — $m$, rise/run, rate of change, $\dfrac{dy}{dx}$, gap, common difference — same number every time.
Polynomial anatomy: coefficient, variable, power (= index = degree), constant; named by terms (monomial / binomial / trinomial) and by degree (linear / quadratic / cubic / quartic).
The number systems $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ tell us which numbers we’re allowed to use in any given question. Most of the time, the unspoken default is $x \in \mathbb{R}$.
From here, every chapter in this block reads as a job done on these objects: factor a polynomial, simplify a fraction made of polynomials, solve an equation for an unknown $x \in \mathbb{R}$. Eight chapters — one vocabulary.
That’s the Algebra Introduction.
One savings story; four ideas that every chapter ahead is built on.