INTRODUCTION
In mathematics, the Highest Common Factors (HCF), also known as the Greatest common factor (GCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the HCF of 8 and 12 is 4.
This notion can be extended to polynomials.
In algebra, the Highest common factor (frequently abbreviated as HCF) of two polynomials is a polynomial, of the highest possible degree, that evenly divides each of the two original polynomials. This concept is analogous to the highest common divisor of two integers.
In algebra, the Highest common divisor (frequently abbreviated as HCF) of two polynomials is a polynomial, of the highest possible degree, that evenly divides each of the two original polynomials. This concept is analogous to the highest common divisor of two integers.
CHAPTER # 1
FACTORIZATION
Factorization mean of splitting a number or polynomial in its components, factors of a number or polynomial are those that divide the number with zero remainder, for example
Factorization of digits
Factors of 9 , 12, 6 and 18
9: factors are 1,3,9
12: factors are 1,2,3,4,6,12
6: factors are 1,2,3,6
18: factors are 1,2,3,6,9,18
Factors of 15 are 1, 3, 5, and 15
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
CHAPTER # 2
FACTORIZATION OF POLINOMIALS
There are different ways of finding the factors of a polynomial depend on the situation some of them are discuss below
by taking common K method e.g.
Kx+Ky+K=K(x+y+1) here K is factor
By head to tail method eg
4x2+7x+3=(x=1)(4x+3)
by squaring formula i.e (a+b)2,(a-b)2 eg:
4x2+4x+1=(2x+1)(2x+1)
4x2-4x+1=(2x-1)(2x-1)
by sum and difference formula i.e a2-b2,a3+b3,a3-b3
a2-b2=(a+b)(a-b) eg:
9x2-16y2=(3x-4y)(3x+4y)
a3+b3=(a+b)(a2-ab+b2) eg:
27x3+8y3=(3x+2y)(9x-6xy+4y)
a3-b3=(a-b)(a2+ab+b2) eg:
27x3-8y3=(3x-2y)(9x+6xy+4y)
by cube formula (a+b)3, (a-b)3
(a+b)3=(a+b)(a+b)(a+b) eg:
(ax+by)3=(ax+by)(ax+by)(ax+by)
(a-b)3 =(a-b)(a-b)(a-b) eg:
(3a-2b)3=(3a-2b)(3a-2b)(3a-2b)
By remainder theorem eg:
x2 + 7x + 6 = (x + 1)(x + 6)
x2 − 5x − 6 = (x + 1)(x − 6)
By formula of the forms (a+b+c)2 and a3+b3+c3-3abc
(a+b+c)2=(a+b+c)(a+b+c)
a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)
CHAPTER # 3
HIGHEST COMMON FACTOR OF DIGITS
The "Highest Common Factor" is the largest of the common factors (of two or more numbers)
Finding the Highest Common Factor
Here are three ways:
• You can:
find all factors of both numbers
then select the ones that are common to both, and
then choose the Highest.
Example:
Two Numbers Factors Common Factors Highest
Common Factor Example Simplified
Fraction: 9 and 12
9: 1,3,9 1,3 3
12: 1,2,3,4,6,12 1,3
9/12 = 3/4 which dived both numbers
And another example:
Two Numbers Factors Common Factors Highest
Common Factor Example Simplified
Fraction 6 and 18
6: 1,2,3,6
18: 1,2,3,6,9,18 1,2,3,6 6
6/18 = 1/3 which dived both numbers
2. You can find the prime factors and combine the common ones together:
Two Numbers Thinking ... Highest
Common Factor Example Simplified
Fraction
24 and 108
2 × 2 × 2 × 3 = 24, and 12
2 × 2 × 3 × 3 × 3 = 108 2 × 2 × 3 = 12
24/108 = 2/9 which dived both numbers
3. And sometimes you can just play around with the factors until you discover it:
Two Numbers Thinking ... Highest
Common Factor Example Simplified
Fraction
9 and 12 3 × 3 = 9 and 3 × 4 = 12 3 3
9/12 = ¾ which dived both numbers
Let's start with an Example...
Highest Common Factor of 12 and 30
1. Find all the Factors of each number,
2. Circle the Common factors,
Factors of 12 are 1, 2, 3, 4, 6 and 12
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
3. Choose the Highest of those
Factor 6 is the Highest of 1,2,3,6
A number can have many factors:
Example: The common factors of 15, 30 and 105
Factors of 15 are 1, 3, 5, and 15
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
The factors that are common to all three numbers are 1, 3, 5 and 15
In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15
What is the "Highest Common Factor”?
It is simply the largest of the common factors.
In our previous example, the largest of the common factors is 15, so the Highest Common Factor of 15, 30 and 105 is 15
