When a student enters high school, mathematics is divided into 2 subjects: algebra and geometry. There are more and more concepts, tasks are becoming more difficult. Some people have difficulty understanding fractions. 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 a definition. Algebraic fraction refers to P/Q expressions, where P is the numerator and Q is the denominator. A number, a numeric expression, a numerical-alphabetic expression can be hidden under an alphabetic entry.
Before you wonder how to solve algebraic fractions, you first need to understand that such an expression is part of a whole.
Usually, an integer is 1. The number in the denominator shows how many parts the unit is divided into. The numerator is needed in order to find out how many elements are taken. The fractional bar corresponds to the division sign. It is allowed to record a fractional expression as a mathematical operation "Division". In this case, the numerator is the dividend, the denominator is the divisor.
Basic rule of common fractions
When students go through this topic at school, they are given examples to reinforce. To solve them correctly and find different ways out of 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 an ordinary fraction will not change. A special case of this rule is the division of both parts of the expression into the same number or polynomial. Such transformations are called identical equalities.
Below, we will discuss how to solve addition and subtraction of algebraic fractions, to perform multiplication, division and reduction of fractions.
Math operations with fractions
Let's consider how to solve the basic property of an algebraic fraction, how to apply it in practice. Whether you need to multiply two fractions, add them, divide one by the other, or subtract, you must always follow the rules.
So, for the operation of addition and subtraction, you should find an additional factor to bring the expressions to a common denominator. If initially the fractions are given with the same expressions Q, then you need to omit this item. When the common denominator is foundsolve algebraic fractions? Add or subtract 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.
The concept of fraction reduction is often used. This means the following: if the numerator and denominator are divided by an expression other than unity (the same for both parts), then a new fraction is obtained. The dividend and divisor are smaller than before, but due to the basic rule of fractions they 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 greatest common divisor. The operation algorithm consists of two items:
- Finding the GCD for both sides of a fraction.
- Dividing the numerator and denominator by the found expression and getting an irreducible fraction equal to the previous one.
The table below shows the formulas. For convenience, you can print it out and carry it with you in a notebook. However, so that in the future when solving a test or exam there will be no difficulties in the question of how to solve algebraic fractions, these formulas must be learned by heart.
Several examples with 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 understandmaterial.
1. Convert fractions and bring them to a common denominator.
2. Convert fractions and bring them to a common denominator.
3. Reduce the given expressions (using the learned basic rule of fractions and reduction of powers)
4. Reduce polynomials. Hint: you need to find the abbreviated multiplication formulas, bring them to the proper form, reduce the same elements.
Assignment to consolidate the material
1. What steps need to be taken to find the hidden number? Solve the examples.
2. Multiply and divide fractions using the basic rule.
After studying the theoretical part and considering the practical issues, no more questions should arise.
