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Algebra Fractions

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

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

Fraction multiplication Rule

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.

Example

Fraction division Rule

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.

Example

Property of Two Fraction

Two fraction are equivalent if and only if their cross products are equal.

Example

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.

Example

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.

Example

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.

Example

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 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

Least Common Denominator (LCD)

Least common denominator is the product of all factors which have the highest exponent in all denominators.

Example

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 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