Terminology

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is the basis of the Leaving Certificate mathematics course as every subsequent chapter requires a sufficient understanding of algebraic topics.

Terminology

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Operations

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Operations are the essence of any algebraic system and mastering their relationship with other operations is crucial in understanding algebra.

Addition and Subtraction

In algebra, only like terms can be added or subtracted.

• $2x$ and $x$ are like terms, they share the same variable $x$ raised to the same power $x^1$.

• $3$ and $7$ are like terms, any constants are like terms of each other.

• $3a^2b$ and $8a^2b$ are like terms, they share same combination of variable to the same power.

• $3(a+b)$ and $-2(a+b)$ are like terms

• $2x$ and -7$$y are not like terms, they don't share the same variable name.

• $x$ and $x^2$ are not like terms, they share the same variable but not raised to the same power.

• $4ab^2$ and $-2ba^2$ are not like terms, they don't share the same combination are like terms.

When adding or subtracting like terms, only sum their coefficients. For example $4x+2x=6x$.

For example, let's simplify the expression :

$2x + 5y - x + 4 - y + x - 1$

First identify all the like terms. There are three $x$ terms. $2x-x+x=2x$

$2x + 5y + 4 - y - 1$

There are two $y$ terms, $5y-y=4y$

$2x + 4y + 4 - 1$

There are two constant terms, $4-1=3$

$2x + 4y + 3$

This is the simplified expression, the simplest form is when no more like terms can be added.

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Addition is commutative, meaning it doesn't matter in which order we arrange addition, for example $6+4$ is the same value as $4+6$.

Subtraction is not commutative, but subtraction can be written in terms of addition. For example, if we wanted to rearrange :

Rewrite in terms of addition (the sign is always attached to the term)

Rearrange using the commutative law of addition.

Remove brackets :

Multiplication

Multiplication does not concern itself with like terms, we can simply follow basic multiplication rules, multiply constants with constants, variables with variables etc.

When multiplying variables, we follow indices rules, when the base is the same but the power is different, add the powers together.

Let's have a look at some examples :

Multiplication has an interesting property when paired with addition of subtraction, the law of distribution.

This law is very powerful because it let's us expand brackets. That is, when we want to multiply two brackets, the first bracket distributes amongst each term in the second bracket.

Now distribute $a$ amongst $b$ and $c$.

Division

Similar to multiplication, we handle respective components, divide coefficients and subtract exponents where applicable. We refer to the indices rules again :

A key property of fraction is that when the numerator and denominator share a common factor, it cancels out :

Let's explore some examples :

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Recall we handle components separately, so the constants divide ($4÷2=2$) and the $x$-terms divide, using indices rules.

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$8$ and $3$ don't divide evenly so we can leave those untouched if we want to avoid decimal notation. The $b$-terms are handled using indices rules and the $a$ term doesn't have anything to cancel out with on the bottom of the fraction, so this stays unaffected.

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Indices

In the same way multiplication represents repeated adding, exponents represent repeated multiplication.

Indices have their own unique properties - two of which we've already seen. We'll explore the rest in a subsequent section.

Let's have a look at an example for now :

This expression says that we repeatedly multiply $(2x-4)$ twice, so let's rewrite it in multiplicative form :

From here we should know what to do, try it as an exercise.

Precedence of Operations

So far we have met the five basic operations of algebra - addition, subtraction, multiplication, division and exponentiation (or indices). Some operations come before others, we can refer to BIMDAS to see which ones to do first.

Let's work through an example :

First we handle brackets - start with the most nested bracket

Nothing to simplify, onto the other bracket :

In here, we can simplify

There's nothing left to simplify in the brackets, we move onto indices :

Next is multiplication :

Addition and subtraction actually have the same precedence, clean up the expression from left to right :