# Mathematics Factoring Basics

Factoring is very important in algebra because it allows us to perform operations on fractions with different denominators, write polynomials in a simpler form, and make many calculations a lot simpler. This article explains the very basics of factoring.

Whole numbers are the natural numbers that includes 0, 1, 2 and so on. Integers are the same as a whole number, but also includes negative numbers. Factoring whole numbers and Integers is the same, except with an integer we may have to deal with a -1 factor.

Integers can be composite numbers, or prime numbers. A composite number has two or more factors. When those factors are multiplied together, the result is the original number. A prime number is an integer that is not a product of two smaller integers. The only factors a prime number has are itself and the number 1.

Prime numbers up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

You can use a "factor tree" to factor an Integer. To create a factor tree, write the number at the top of your paper. Draw two short diagonal lines from the bottom of the number, one slanted to the left and the other slanted to the right. Just below each slanted line write a factor of the number.

**•** If there are no factors that equal the original number when
multiplied together (other than that number and 1) then the number is a
prime number and cannot be made into a factor tree.

Shown below is a facator tree for the number 15:

15 | ||||

/ | \ | |||

3 | 5 |

When 3 and 5 are multiplied together they equal 15, therefore they are factors of 15. Because 3 and 5 are prime numbers they can not be factored further. They are prime factors.

Shown below is a factor tree for the number 27:

27 | ||||

/ | \ | |||

3 | 9 |

When 3 and 9 are multiplied together they equal 27, therefore they are factors of 27. 3 is a prime factor and so can not be factored further. However 9 can be factored into 3 and 3, so the tree grows as shown below.

27 | |||||||

/ | \ | ||||||

3 | 9 | ||||||

/ | \ | ||||||

3 | 3 |

Now, 27 has been completely factored into prime factors so can not be factored further.

Shown below is a factor tree for the number 24:

24 | ||||

/ | \ | |||

2 | 12 |

When 2 and 12 are multiplied together they equal 24, therefore they are factors of 27. 2 is a prime factor and so can not be factored further. However 12 can be factored into 2 and 6, so the tree grows as shown below.

24 | ||||||||

/ | \ | |||||||

2 | 12 | |||||||

/ | \ | |||||||

2 | 6 | |||||||

/ | \ | |||||||

2 | 3 |

12 can also be factored into 3 and 4, making the tree grow as shown below.

24 | ||||||||

/ | \ | |||||||

2 | 12 | |||||||

/ | \ | |||||||

3 | 4 | |||||||

/ | \ | |||||||

2 | 2 |

24 can also be factored into 3 and 8, making the tree grow as shown below.

24 | ||||||||

/ | \ | |||||||

3 | 8 | |||||||

/ | \ | |||||||

2 | 4 | |||||||

/ | \ | |||||||

2 | 2 |

To list all the factors of 24, in order of value, list each factor from the trees above one time, as shown below.

2, 3, 4, 6, 8, 12

Note the list includes factors that are not prime numbers. When choosing factors for a number, always start with the lowest prime number that can be divided into the number. That way at the end, you are also determining the numbers prime factors.

**Greatest Common Factor (GCF)**

4x^{2} - 20x + 16

The polynomial shown above can be simplified by dividing each term's coefficient by the GCF. Using a factor tree to factor each coefficient reveals that the GCF 4. The simplified polynomial is shown below.

4(x^{2} - 5x + 4)

**Lowest Common Multiple (LCM)**

LCM is used for adding or subtracting fractions where denominators are not same. An example is shown below:

7 12 | + | 5 18 |

As described above, by using the lowest prime number that can be divided into the number you are also determining the numbers prime factors.

prime factors of 12: 2, 2, 3.

Prime factors of 18: 2, 3, 3.

Then multiply together the highest multiples of prime factors of 12 and 18:

2 x 2 x 3 x 3 = 36

This gives you the LCM. Convert both fractions to fractions with the LCM in their denominators.

7 12 | x | 3 3 | = | 21 36 | |||

5 18 | x | 2 2 | = | 10 36 |

Now you can add the two fractions.

21 36 | + | 10 36 | = | 31 36 |