What are Linear Expressions?
Any algebraic expression is considered to be a linear expression when it has terms containing variables with exponent equal to one.
For example:
2x + 3y + 3
Observe that this expression contains two variables, x and y.
There is just one constant, 3 in the expression.
Also there are two coefficients, 2 and 3, seen in the terms, 2x and 3y.
One important observation in the expression given here is that the exponent of the variables in the terms is one, and hence, the expression is a linear expression.
Defining Monomial, Binomial and Polynomial Linear Expressions
Linear expressions in algebra are classified as monomials, binomials and trinomials based on the number of terms in the expression. In general, the word polynomial is used to describe linear expressions with more than two terms.
Examples:
Monomial: 3xy, 4a, -ab, 16tu, etc. – shows just one term in the expression.
Binomial: 3jk-5gh, 5x-3ef, 13cd+4xyz, etc. – shows that there are two terms in each expression.
Trinomial: 43+4fgh + 5ef, 7tu+56+gh, etc. – shows that there are three terms in each expression.
Define factoring of Linear Expressions.
Factoring in math refers to breaking up a number or an algebraic expression into a form that shows the number or the expression as a product of numbers or algebraic terms.
For example, in the case of a number, 48, we break this number into its factors that are, 1, 2, 3, 4, 6, 8, 12, 16, 24, 48, a group of ten factors.
On the other hand, algebraic expressions with variables, constants and coefficients are solved in terms of finding the GCF. Let’s factor a simple trinomial linear expression, 4x+16xy+16.
By finding the GCF of the expression, we can factor this expression, and this is written as, 4(x+4xy+4). Observe that 4 is the GCF for all the terms of the expression.
Applying the Area Model to Factor Expressions
The factoring of algebraic expressions can also be done with the help of a table in the following manner.
Consider a binomial linear expression, 3x+ 9
This is also written as, 3(x) + 3(3) = 3 (x+3)
We must also note that 3(x) + 3(3) represents the distributive property of addition.
Rapid Recall
In general, factoring linear algebraic expressions with more than two terms is done in the following way:
Solved Examples
Example 1:
Factor the expressions given in the list by using the GCF method.
- 4x+24y
- 3st+9sv+12sx
- 8m -32m-24
Solution:
1. 4x + 24y
4 (x) + 4(6y) [Write the expression using the GCF]
4(x+ 6y) [Write expression using the distributive property]
2. 3st+9sv+12sx
3s(t)+ 3s(3v) + 3s(4x) [Write the expression using the GCF]
3s(t+3v+4x) [Write expression using the distributive property]
3. 8mn -32pm-24
8(mn)- 8(4pm) – 8(3) [Write the expression using the GCF]
8(mn-4pm-3) [Write expression using distributive property]
Example 2:
State whether the expressions in the options have been factored in the right manner.
- 6x + 2y + 2z = 2(3x+y+z)
- 4mt – 8mn – 16ms = 2(2mt-4mn-8ms)
- 10ef- 5en + 15 = 5(2ef -en +3)
Solution:
- In this option, 6x + 2y + 2z = 2(3x+y+z), the expression has been factored correctly as ‘2’ is the GCF of the terms in the expression.
- In this option,4mt – 8mn – 16ms = 2(2mt-4mn-8ms),the factoring has been done incorrectly as the GCF of the terms in the expression is ‘4’ and not ‘2’.
Hence, the correct factoring is,4(mt-2mn-4ms).
3. In this option, 10ef- 5en + 15 = 5(2ef -en +3), the expression has been factored correctly as ‘5’ is the GCF of the terms in the expression.
Example 3:
A truck has to load eight cartons of clothes into Warehouse A. Each of the cartons weigh ‘r’ pounds. If some of the clothes are distributed to other warehouses on the way to Warehouse A, and the eight cartons are now lighter, what is the weight of the clothes per carton distributed in Warehouse A? The cartons weighed 8r – 16 on reaching Warehouse A.
Solution:
To find the weight of clothes per carton on reaching Warehouse A factor the expression 8r – 16.
8r – 16 [Write expression]
8(r)-8(2) [Use GCF]
8(r-2) [Apply the Distributive Property]
Hence each of the cartons weighs (r-2) pounds when it arrived at Warehouse A.
Each part of an algebraic expression that is separated by addition or subtraction operation, containing variables, coefficients and constants, is known as a ‘term’.
A non-linear algebraic expression is different from a linear algebraic expression as the exponent of the terms in non-linear expressions is either 2 or greater than 2.
Combining like terms is a method that is used to solve algebraic expressions with multiple terms. Combining terms with the same variables helps in solving the expression quickly.