In the previous lesson, you learned about distributive law and how to use them to rewrite expressions. The distribution law applied on factored expression and it changes into distributed form. The factored form of expression is showing the function as a product of two simpler expressions which when multiplied will be in expanded form.
In this article, I am going to talk about factors and factoring expressions. I assume that you know about expression, if not then you can read previous lessons of this course.
Factors
All numbers are expressions, called numeric expressions or simply constants. There are two types of numeric constants – prime numbers and composite numbers. The prime numbers have only two factors – 1 and itself. If a is prime number , then its factors are 1 and a. Other numbers which are composite numbers, have more factors including 1 and the number itself.
What are factors of a number ?
To understand factors let’s take an example number – 12.
Let’s make a list of all numbers that divide 12 perfectly which means if you divide 12 by those numbers you selected , you get 0 as remainder. I have listed then below.
1, 12, 6, 2, 4, 3
These are factors of number 12. If we multiply these numbers ,we get 12. For example, 12 and 1, 6 and 2, 4 and 3 multiply to give us 12.
Let us now try to find factors of a prime number. For example, we can find factors of number 7. You will note that number 7 has no other divisor than 1 and itself , therefore, its factors are 1 and 7. This is a prime number.
Prime Factorization
Every number is multiple of prime numbers and the number 1 is factor of all numbers. To find all prime numbers that are factors of a number is called prime factorization.
For example,
We can divide number 12 by 2 which is prime factor. We get 6 as quotient, the number 6 is divisible by 2 , which gives 3 as quotient. We divide 3 by 3 and the quotient is 1. Therefore, by prime factorization, our factors are 
or you can write 
. We only count prime numbers and leave 1 and 12 which are also factors of 12.
Greatest Common Factor
Sometimes you want to compare two numbers and find a greatest factor that are common to both number. It is called greatest common factor or GCF. We can find GCF of two numbers by listing all known factors of a number or by prime factorization method.
For example,
To find the greatest common factor of number 10 and 25 using the factor listing method.
List all factors of 10 : which is: 1, 10, 2, 5
List all factors of 25: which is: 1, 25, 5
The number 5 is the greatest common number and hence, it is the GCF of 10 and 25. You can find the GCF of more than two numbers without any problem using the same method.
Let us, find the GCF through prime factoriszation method.
We take two number which is 10 and 25 again.
If we divide 10 with prime numbers only, we get.
5 and 2 as prime factors.
If we divide 25 with prime numbers only we get,
5 because 25 is square of 5.
Once again the common factor is 5 which is the greatest common factor for 10 and 25.
Factoring expressions
An expression is made of terms and each term is made of coefficient and variables. By taking common factors, we are breaking the expression into simple expressions and show it as product of those simple expressions. Factors multiply to give the expanded form of expression. So we separate greatest common factors from each term, the negative sign, and least common multiples of variables during the process of factorization.
Steps to find the the factors of an expression is:
- Find the greatest common factor of coefficents of expression.
- Find out if sign of factor is negative or positive.
- Least common multiples of variables
- Write down the factors as product of simple expressions.
Let us see few examples to understand this better. But, before that you must understand few terms about expressions.
An expression with single term is called a monomial. for example,
3x^2 is a monomial
5b is a monomial.
expression with two terms is called a binomial and expression with three term is called a trinomail. An expressiion made of several monomials is called a polynomial. All expression with one or more terms is essentially a polynomial.
Example 1:
Write the following monimial in factored form:
6x^3
The first step is to find the GCF for the coefficient 6.
We use prime factorization method and list out all prime factors for 6.
2, 3
second step is to, check if the term is negative, if yes, then our factor is negative. otherwise , it is positive only.Alway check the first term only. The first term decides whether GCF is a positive or a negative number.
third step, find the variables and their multiples.
here we have x raised to 3 which means variable ‘x’ multipled 3 times.
x . x . x
We can write out expression in factored form as
(2x^2) multiplied by (3x) and if you multiply them, you get 6x^3.
Example #2
Write the following expression in factored form:
-6x^3 + 21x^2
First step is to find the factors of coefficiets of both terms using prime factorization method.
4 has factor 2,3
21 has factors 3,7
The greatest common factor is 3.
Step 2 is to check the sign, the greatest common factor is negative because the first term of the expression is negative.
Step 3 is find the least common multiple for variables.
first term has x^3 = x multiplied 3 times.
second term has x^2 = x multipled by x
x multipled x is least common multiple.
therefore, our factors are -3x^2 multipled by 2x – 7.
We must be careful about signs because negative multipled by negative is positive value.
Summery
Let summerize that you learned in this lesson. To find the factors of an expression, you must do the following
- Find the GCF or greatest common factor for all terms in the expression.
- If the first term is negative then the factor is also negative.
- List out the least common multiple for variables.
- Write down your expression as a product of simple expressions.
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Thank you.
