6.2 Algebraic notation
An algebraic expression indicates a sequence of calculations that can also be described in words or by means of a flow diagram, as you have seen in the previous chapter.
For example, this is how an algebraic expression is constructed:
- in words: “multiply a number by two and then add six”
- using flow diagram:
Remember, the flow diagram illustrates the order in which the calculations must be done.
- using algebraic notation: \(x \times 2 + 6\) or \(2x + 6\)
In algebraic language, the multiplication sign is usually left out. So we can write \(2x\) instead of \(2 \times x\). We also write \(x \times 2\) as \(2x\).
In an algebraic expression, the unknowns are called variables. They stand in the place of numbers until we decide which numbers we want to put in their place. For example, given the expression \(3x - 5y\), the variables are \(x\) and \(y\).
- variable
- a symbol used to represent a number; usually we use letters for variables, but they can be anything other than numbers
Worked example 6.1: Algebraic expressions from words
Write the algebraic expression that matches the words.
“The difference when \(r\) is subtracted from \(p\).”
Identify the key words.
“Difference” is what is left after we do subtraction. We must subtract in the given order.
Determine the difference.
\(p\) and \(r\) are the variables. The question says that \(r\) is subtracted from \(p\). We must start with \(p\) and subtract \(r\).
So, the difference when \(r\) is subtracted from \(p\) is \(p - r\).
Hint: We cannot simplify \(p - r\) any further, because \(p\) and \(r\) are not like terms. \(p - r\) is the best we can get. In algebra, the answer is often an expression instead of a single number! This is something that you will have to get used to.
Worked example 6.2: Algebraic expressions from words
Write the algebraic expression that matches the words.
“The product of \(p\) and \(r\).”
Identify the key words.
“Product” is what we get after we do multiplication. We can multiply in any order. For example, the product of 3 and 4 is \(3 \times 4 = 4 \times 3 = 12\).
Determine the product.
So, the product of \(p\) and \(r\) is: \(p \times r = pr\).
Hint: In algebra, we don’t usually write the times sign (\(×\)). It’s not incorrect to write it – it just takes up extra space. Instead, we just write the variables next to each other, with nothing in between.
Look at the expression, \(3x^{2} + 6\). Each part of the expression has a special name:
| Coefficient | Variable | Exponent | Constant |
|---|---|---|---|
| 3 | 𝑥 | 2 | 6 |
- coefficient
- a number that is multiplied by a variable; it is usually in front of the variable
- constant
- a number (without a variable) that is added or subtracted
- exponent
- a number of variable that shows us how many factors there are
In an expression that can be written as a sum, the different parts of the expression are called the terms of the expression. For example, \(3x\), \(2z\) and \(y\) are the terms of the expression, \(3x + 2z + y\).
An expression can have like terms or unlike terms, or both.
- like terms
- terms that have the same variable(s) raised to the same power
- unlike terms
- terms that do not have the same variable(s) raised to the same power
For example, the terms \(3x\) and \(2x\) are examples of like terms, but the terms \(2x\) and \(2z\) are unlike terms.
Terms are like terms if two things are true:
- the terms must have the same variable or variables (or both terms must have no variable)
- the variables must have the same exponents.
Another way to summarise this is: like terms must have the same number of variable factors. (Notice that the coefficients of the terms do not have anything to do with whether the two terms are like terms or not.)
Worked example 6.3: Like terms
You are given two terms: \(- 4k^{3}\) and \(- 2k^{2}\).
Are these like terms or not?
Compare the variables.
Yes, both \(- 4k^{3}\) and \(- 2k^{2}\) have the same variable, \(k\).
Compare the exponents.
Now we must also check the exponents of the variables: are they equal? No, the terms have different exponents: the first exponent is \(3\) and the second is \(2\).
Draw your conclusion.
The terms, \(- 4k^{3}\) and \(- 2k^{2}\), are not like terms because they have different exponents.
Worked example 6.4: Like terms
You are given two terms: \(- 4m\) and \(- 2\).
Are these like terms or not?
Compare the variables.
No, \(- 4m\) and \(- 2\) do not have the same variable. In fact, the second term has no variable at all, so \(- 2\) is a constant term.
Draw your conclusion.
The terms, \(- 4m\) and \(- 2\), are not like terms.
Worked example 6.5: Like terms
You are given two terms: \(3b^{2}\) and \(- 6b^{2}\).
Are these like terms or not?
Compare the variables.
Yes, both \(3b^{2}\) and \(- 6b^{2}\) have the same variable, \(b\).
Compare the exponents.
Now we must also check the exponents on the variables: are they equal? Yes, the terms have the same exponents – the first exponent is \(2\) and the second is also \(2\).
Draw your conclusion.
The terms, \(3b^{2}\) and \(- 6b^{2}\), are like terms because they have the same variables and the same exponents.