When a student goes to high school, mathematics is divided into 2 subjects: algebra and geometry. There are more and more concepts, tasks more and more complicated. Some have difficulty perceiving fractions. We missed the first lesson on this topic, and voila. How to solve algebraic fractions? A question that will torment throughout school life.
The concept of algebraic fraction
Let's start with the definition. By an algebraic fraction we mean the expressions P / Q, where P is the numerator and Q the denominator. A number, a numerical expression, a numerical-alphanumeric expression may be hidden under the letter entry.
Before you ask yourself how to solve algebraic fractions, you first need to understand that such an expression is part of the whole.
As a rule, the integer is 1. The number in the denominator shows how many parts the unit was divided. The numerator is needed in order to find out how many elements are taken. The fractional line corresponds to the division sign. It is allowed to record a fractional expression as a mathematical operation "Division". In this case, the numerator is a dividend, the denominator is a divisor.
The basic rule of ordinary fractions
When students complete this topic at school, they are given examples of reinforcement. In order to solve them correctly and find different ways from difficult situations, you need to apply the basic property of fractions.
It sounds like this: If you multiply both the numerator and the denominator by the same number or expression (other than zero), then the value of the ordinary fraction will not change. A special case of this rule is the separation of both parts of the expression into the same number or polynomial. Such transformations are called identity equalities.
Below we will consider how to solve the addition and subtraction of algebraic fractions, to produce the multiplication, division and reduction of fractions.
Mathematical operations with fractions
Consider how to solve the basic property of an algebraic fraction, how to apply it in practice. If you need to multiply two fractions, add them, divide one into another or subtract, you must always adhere to the rules.
So, for the addition and subtraction operation, an additional factor should be found to bring the expressions to a common denominator. If initially fractions are given with the same expressions Q, then this item should be omitted. When a common denominator is found, how to solve algebraic fractions? You need to add or subtract the numerators. But! It must be remembered that if there is a “-” sign in front of the fraction, all signs in the numerator are reversed. Sometimes you should not perform any substitutions and mathematical operations. It is enough to change the sign before the fraction.
Often used the concept of reduction of fractions . This means the following: if the numerator and denominator are divided into an expression different from unity (the same for both parts), a new fraction is obtained. The dividend and divisor are smaller than the previous ones, but by the basic rule of fractions remain equal to the original example.
The purpose of this operation is to obtain a new irreducible expression. This problem can be solved by reducing the numerator and denominator by the largest common factor. The operation algorithm consists of two points:
- Finding GCD for both parts of the fraction.
- Dividing the numerator and denominator by the expression found and obtaining an irreducible fraction equal to the previous one.
The table below shows the formulas. For convenience, it can be printed and carried with you in a notebook. However, so that in the future when solving the control or exam there are no difficulties in the question of how to solve algebraic fractions, these formulas must be learned by heart.
Some examples of solutions
From a theoretical point of view, the question of how to solve algebraic fractions is considered. The examples in this article will help you better understand the material.
1. Convert fractions and bring them to a common denominator.
2. Convert fractions and bring them to a common denominator.
3. To reduce the indicated expressions (using the studied basic rule of fractions and reduction of degrees)
4. Reduce polynomials. Hint: you need to find formulas of abbreviated multiplication, lead to a proper form, reduce the same elements.
Assignment task
1. What actions need to be performed to find a hidden number? Solve the examples.
2. Multiply and divide fractions using the basic rule.
After studying the theoretical part and examining practical issues should no longer arise.