Introduction
When writing a numeric expression we use numbers and basic operations to build up a number sentence that can be later calculated. Algebraic expressions are the same as numeric expressions except that they also use some new algebraic tools. These new algebraic tools are pronumerals and coefficients.
Pronumerals and coefficients
We use algebraic expressions when we want to write a number sentence but don't know all the numbers involved.
For example: What is the total weight of a cat and a dog?
This is an algebraic expression as it is a number sentence that uses pronumerals in the place of some numbers.
A pronumeral is a symbol, commonly a letter, that is used in the place of a numeric value.
Don't use mathematical operation symbols as pronumerals otherwise things won't make sense.
For example: '+-\lt = \%' could be an algebraic expression, but it is also very confusing.
Coefficients are used in algebraic expressions to represent how many sets of a pronumeral we have. They are written in front of a pronumeral without a multiplication symbol like so: 3u=3 \times u The pronumeral is u with a coefficient of 3.
Notice how we don't need the multiplication symbol to represent multiple sets of a pronumeral. This is because there is no danger of mixing up a coefficient next to a pronumeral with any other term, whereas if we did this for numbers they would get mixed up with two digit numbers (for example: 3\times 4=12 \neq 34)
A coefficient is a numeral that is placed before and multiplies a pronumeral in an algebraic term.
Coefficients are a bit different from multiplication though, since they also include the sign of the term. This means that a negative term, -6q for example, has a coefficient of -6.
We can see this more clearly in a longer expression.
Consider the expression: 4x-3y+7z
By breaking up the expression into its individual terms we can determine the coefficients of each pronumeral.
| Term | Coefficient | Pronumeral |
|---|---|---|
| 4x | 4 | x |
| -3y | -3 | y |
| +7z | 7 | z |
We can also have algebraic terms where the coefficient is a fraction.
Consider: v\div 4 = \dfrac{v}{4}=\dfrac{1}{4}\times v
Since dividing by a number is the same as multiplying by its reciprocal , dividing by 4 gives us a coefficient of \dfrac{1}{4}.
What about variables that don't appear to have coefficients?
Consider the term x.
Since x is equal to 1 \times x which is also equal to 1x, it actually has a coefficient of 1. Whenever a pronumeral has no written coefficient, its coefficient can be assumed to be 1.
Similarly, the coefficient of -x is -1.
Examples
Example 1
Find the coefficient of y in:
4y
-4y
A coefficient is a number that is placed before and multiplies a pronumeral in an algebraic term.
Basic operations in algebra
Aside from the use of coefficients in multiplication, the basic operations work almost the same for algebraic terms as they do for numbers.
Between pronumerals and numbers we have:
| Word expression | Algebraic expression | Simplified algebraic expression |
|---|---|---|
| \text{Three more than}\,x | x+3 | x+3 |
| \text{Three less than}\,x | x-3 | x-3 |
| \text{The quotient of }\ x\, \text{and three} | x\div 3 | \dfrac{x}{3} |
| \text{The product of }\ x\, \text{and three} | x\times 3 | 3x |
Between pronumerals and other pronumerals we have:
| Word expression | Algebraic expression | Simplified algebraic expression |
|---|---|---|
| y\,\text{more than}\,x | x+y | x+y |
| y\,\text{less than}\,x | x-y | x-y |
| \text{The quotient of }\ x\,\text{and}\, y | x\div y | \dfrac{x}{y} |
| \text{The product of }\ x\,\text{and}\, y | x\times y | xy |
As we can see from the tables, addition and subtraction in algebraic expressions does not usually simplify.
The only time they will simplify is when they are like terms .
It should also be noted that the division doesn't actually simplify but is instead written as a fraction, which is slightly more compact and removes the need to use brackets in more complicated expressions, for example: 4\div \left(x+3\right)=\frac{4}{x+3}
Technically, the multiplication between a pronumeral and a number also uses a more compact form by removing the multiplication symbol, but in this case using a coefficient is considered simplifying.
The same can be said for the multiplication between different pronumerals except, in this case, there is no coefficient and instead we have two pronumerals.
These operations will work the same way when applying more than one of them.
Examples
Example 2
What does the expression 8x mean?
Example 3
Write an algebraic expression for the following phrase "eight more than the quotient of 9 and x".
Operations with pronumerals:
| Word expression | Algebraic expression | Simplified algebraic expression |
|---|---|---|
| \text{Three more than}\,x | x+3 | x+3 |
| \text{Three less than}\,x | x-3 | x-3 |
| \text{The quotient of }\ x\, \text{and three} | x\div 3 | \dfrac{x}{3} |
| \text{The product of }\ x\, \text{and three} | x\times 3 | 3x |
| y\,\text{more than}\,x | x+y | x+y |
| y\,\text{less than}\,x | x-y | x-y |
| \text{The quotient of }\ x\,\text{and}\, y | x\div y | \dfrac{x}{y} |
| \text{The product of }\ x\,\text{and}\, y | x\times y | xy |