Algebra Fractions

Algebra Fraction
If A, and B are polynomials with B is not zero, we define the A/B is an algebraic fraction, in which A is the numerator and B is the denominator.
Example
fractions example
Undefined Fraction
If the denominator B is zero, then the fraction of A/B is undefined. If the numerator A is zero, then A/B = 0. We assume that the variables in any fraction may not be assigned values that will result in a value of zero for the denominator.
Example
undefined fractions example
Fundamental Principle of Fraction
The numerator and denominator of a given fraction multiply the same nonzero monomial or polynomial, the result fraction will be the same as the given original fraction. M is monomial or polynomial.
numerator and denominator of a fraction multiply the same nonzero monomial
Example
Fundamental Principle of Fraction
The numerator and denominator of a given fraction are divided by the same nonzero monomial or polynomial, the result fraction will be the same as the given original fraction. M is monomial or polynomial.
numerator and denominator of a fraction are divided by the same nonzero monomial
Example
fraction example
Property of Two Fraction
Two fraction are equivalent if and only if their cross products are equal.
Two fraction are equivalent if and only if their cross products are equal
Example
fraction examples
Rules of The Signs of Fractions
Fraction has three signs associated with it, the sign of the numerator, the sign of the denominator, and the sign of the fraction. The fraction remain the same if two of these signs have changed.
rules of the signs of fractions
Example
rules of the signs of fractions examples
Multiplication of Fractions
The product of two fractions A/B and C/D is a fraction whose numerator is the product of the two given numerators and whose denominator is the product of the two given denominators.
multiplication of fractions
Example
multiplication of fractions examples
Division of Fractions
When divide one fraction by another, first change the division to multiplication by inverting the divisor, and then divide out the common factors.
division of fractions
Example
division of fractions example
Addition of Fractions with the same denominators
Addition of Fractions with the same denominators
To add fractions with the same denominators, add the numerator and keep the same denominator.
Note, Look the example shown below, when add a fraction, the first step is to make the same denominator, the denominator of the first term is -x - 1 = - (x + 1), and x + 1 is the denominator of the second term. Also note x2 - 1 = (x + 1)(x - 1), here the symbol x2 represent x square. 
Example
Addition of Fractions with the same denominators example
Addition of Fractions with the different denominators
Addition of Fractions with the different denominators
To add fractions with different denominators, first change the fractions to equivalent fractions with the same denominators, and then add the numerators and keep the same denominator.
Example
Addition of Fractions with the different denominators example
Least Common Denominator (LCD)
Least common denominator is the product of all factors which have the highest exponent in all denominators.
Least Common Denominator (LCD)
Example
Least Common Denominator (LCD) example
Subtraction of fractions with the same denominators
Subtraction of fractions with the same denominators
To subtract fractions with the same denominators, subtract the numerator and keep the same denominator. Note: Look the example shown below, in the demoninator of the second term, (1 - x) = - (x - 1), then -[-(x - 1)] = -[-x + 1]= x - 1. Thus, both first term and second term have the same denominator.
Example
Subtraction of Fractions with the same denominators example
Subtraction of fractions with the different denominators
Subtraction of fractions with the different denominators
To subtract fractions with different denominators, first change the fractions to equivalent fractions with the same denominators, and then subtract the numerators and keep the same denominator.
Note: Look the example in the right side, since x2 - 1 = (x + 1)(x - 1). So both numerator and denominator in the first term should multiply (x + 1), and both numerator and denominator in the second term should multiply (x - 1), then we have the same denominator. Then remove parenthesis -(x - 1) = -x + 1.
Example
Subtraction of fractions with the different denominators example