Denominator of arithmetic fraction of a/b call number b showing the sizes of shares of unit of which the fraction is made. A denominator of algebraic fraction of A/B call algebraic expression of B. For performance of arithmetic actions with fractions they need to be given to the smallest common denominator.
It is required to you
- For work with algebraic fractions when finding the smallest common denominator it is necessary to know methods of polynomial factoring.
Instruction
1. Let's consider reduction to the smallest common denominator of two arithmetic fractions of n/m and s/t where n, m, s, t are integers. It is clear, that these two fractions can be given to any denominator which is divided into m and into t. But usually try to lead to the smallest common denominator. It is equal to a least common multiple of denominators of m and t of these fractions. The Least Common Multiple (LCM) of numbers is the smallest positive number which is divided at the same time into all set numbers. I.e. it is necessary to find a least common multiple of numbers m and t in our case. It is designated as NOC (m, t). Further fractions are multiplied by the corresponding multipliers: (n/m) * (NOC (m, t) / m), (s/t) * (NOC (m, t) / t).
2. Let's give an example of finding of the smallest common denominator of three fractions: 4/5, 7/8, 11/14. For a start we will spread out denominators 5, 8, 14 to multipliers: 5 = 1 * 5, 8 = 2 * 2 * 2 = 2^3, 14 = 2 * 7. Further we calculate NOC (5, 8, 14), multiplying all numbers entering at least one of decomposition. NOC (5, 8, 14) = 5 * 2^3 * 7 = 280. Let's notice that if the multiplier meets in decomposition of several numbers (a multiplier 2 in decomposition of denominators 8 and 14), then we take a multiplier more (2^3 in our case). So, the smallest common denominator of fractions is received. It is equal 280 = 5 * 56 = 8 * 35 = 14 * 20. Here we receive numbers by which it is necessary to increase fractions with the corresponding denominators to lead them to the smallest common denominator. We receive 4/5 = 56 * (4/5) = 224/280, 7/8 = 35 * (7/8) = 245/280, 11/14 = 20 * (11/14) = 220/280.
3. Reduction to the smallest common denominator of algebraic fractions is carried out by analogy with arithmetic fractions. For descriptive reasons we will consider a task on an example. Let two fractions (2 * x) / (9 * y^2 + 6 * y + 1) and (x^2 + 1) / be given (3 * y^2 + 4 * y + 1). Let's factorize both denominators. Let's notice that the denominator of the first fraction represents a full square: 9 * y^2 + 6 * y + 1 = (3 * y + 1) ^2. On multipliers it is necessary to apply a method of group to decomposition of the second denominator: 3 * y^2 + 4 * y + 1 = (3 * y + 1) * y + 3 * y + 1 = (3 * y + 1) * (y + 1). Thus the smallest common denominator is equal (y + 1) * (3 * y + 1) ^2. We multiply the first fraction by y polynomial + 1, and the second fraction on a polynomial 3 * y + 1. We receive the fractions provided to the smallest common denominator: 2 * x * (y + 1) / (y + 1) * (3 * y + 1) ^2 and (x^2 + 1) * (3 * y + 1) / (y + 1) * (3 * y + 1) ^2.