Definition and examples of rational fractions
Rational fractions are studied in 8th grade algebra lessons. We will use the definition of a rational fraction, which is given in the algebra textbook for 8th grade by Yu. N. Makarychev et al.
Definition.
Rational fraction
called a fraction whose numerator and denominator are polynomials with natural, integer or rational coefficients.
This definition does not specify whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of the standard form or not. Therefore, we will assume that the notations for rational fractions can contain both standard and non-standard polynomials.
Here are some examples of rational fractions
. So, x/8 and are rational fractions. And fractions do not fit the stated definition of a rational fraction, since in the first of them there is not a polynomial in the numerator, and in the second both in the numerator and in the denominator there are expressions that are not polynomials.
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Abbreviating rational expressions
Having learned which expressions are rational, we will begin to study their transformations. Let us recall the main property of a fraction:
It means that the numerator and denominator can be multiplied by an arbitrary number (except zero), then the value of the fraction will remain the same:
This rule remains true even when variables are used instead of numbers.
For example, the following transformations of rational expressions are possible:
For example, let's say we need to give the fraction
to the denominator 6a2b2.
What exactly is the multiplier denominator needed to get the monomial 6a2b2? It's obvious that
6a2b2 = 2a2b•3b
Therefore, the expressions above and below the fractional line must be multiplied by 3b:
The 3b multiplier we used is called the additional multiplier.
The inverse operation, in which identical factors are removed from the denominator and numerator, is called fraction reduction:
This identity means that fractions can be reduced by removing a common factor, for example:
Similar actions can be performed not only with numerical fractions, but also with fractional expressions:
In the last example, we put the common factors out of brackets (2x and 7y) so that the same sum x + 3y appears above and below the line, which can be reduced.
However, when reducing fractions, it is important to take into account the range of its permissible values, because due to a change in the denominator it may change. For example, let's say you want to plot a function
The numerator contains the difference of squares, which can be factorized:
It would seem that we have obtained a linear function
y = x + 2
whose graph we know is a straight line. But it is defined for all possible x, while the original fraction is meaningless at x = 2, because then the denominator becomes zero. Therefore, the graph of the function will look like a straight line, but one of its points, with coordinates (2; 4), will be a “punctured” point, and excluded:
This figure means that the graph of the function is a straight line, except for the point (2; 4)
The punctured point on the graph is depicted as a small open circle.
The next important property of a fraction is related to the minus sign. The sign in front of the fraction can be transferred either to the denominator or to the numerator:
We also recall that you can swap the minuend and subtrahend in parentheses if you change the sign in front of it:
(a – b) = – (b– a)
Applying these rules allows you to simplify some fractions, for example:
More complex example:
Let's consider such a concept as a homogeneous polynomial. This is the name given to a polynomial in which all monomials have the same degree.
More information about the degree of a monomial can be found in. In short, the degree of a monomial is the sum of the degrees of all variables included in its letter part. For example, the following monomials have degree 4:
- 3x4 (the only variable has a degree of 4);
- 8x3y (the power of y x is 3 and the power of y is 1, 3 + 1 = 4);
- 5x2y2 (the degrees of both variables are 2, 2 + 2 = 4);
- 10у4 (in the literal part there is only the variable y , whose degree is 4).
Accordingly, the polynomial 3x4 + 8x3y + 5x2y2 + 10y4, composed of all these monomials, will be homogeneous. Examples of homogeneous polynomials are also:
- z6 + v6 – 2z2v4 (here the powers of the monomials are 6);
- a2 – ab (the degree of monomials is 2).
For homogeneous polynomials consisting of two variables, a special technique can be used. It is enough to divide it by one of the variables to the power of a polynomial, and you will get an expression that depends on only one fraction. Let's explain this with an example. Suppose we need to calculate the value of the ratio
if another relationship is known:
In the original fraction, it is the ratio of two homogeneous polynomials of the third degree. Therefore, we divide them by y3 (we could also divide them by x3). In this case, the value of the fraction will not change, because we divide the numerator and denominator by the same monomial:
We have received an expression that depends only on the ratio
Let's try to find this value from the condition
It follows that
Now let’s substitute the found relation into formula (1):
Previously, we looked at examples of fractional expressions consisting of polynomials with integer coefficients. If fractional numbers are used, then you can always get rid of them by multiplying the fraction by some number.
For example, given the fraction
The coefficients for y and y2 are fractional. Let's get rid of them. To do this, we use an additional multiplier of 12:
Next, we'll look at adding and subtracting fractional expressions. The easiest way to carry out this operation is when the denominators of the fractions are the same. In such a situation, the rules we already know are used:
Let's add two values:
Their denominator has the same polynomial, and therefore the operation will look like this:
Here we used the squared difference formula in the numerator.
Now let's subtract from the expression
fraction
They have the same denominators, so there are no problems with subtraction:
Note that usually they try to reduce fractional expressions until an irreducible fraction is obtained.
If fractions have different denominators, then they are reduced to a common denominator by multiplying them by some additional factor. Consider the following example:
The denominators of the fractions are different, however, both fractions can be reduced to the denominator 24x2y3. Why to him? The fact is that for the coefficients of the monomials 6x2y and 8x3 the least common multiple (LCM) is the number 24 (you can learn about the LCM from). Let's add to this coefficient the variables from the monomials with the largest exponents (x2 and y3) and get the monomial 24x2y3. So, multiply the first fraction by 4y2, and the second by 3x:
There is an easier way to find the common denominator; to do this, simply multiply the denominators of the fractions. However, further transformations will take longer. Let's solve the same example this way:
In the numerator it is possible to take the common factor 2xy out of brackets:
It can be seen that the final result of the operation has not changed.
If the denominators of the fractions being added contain polynomials, then it is worth trying to factor them. Due to this, sometimes it is possible to find a simpler common denominator.
Suppose we need to add expressions
Let's take the factors x and y out of the denominators:
The denominators have similar factors, (3x – y) and (y – 3x). To make them the same, you need to swap the places of the subtrahend and minuend in the same brackets. To do this, you need to add a minus sign in front of them:
The common factor of these fractions is the product xy(3x – y):
It remains to expand the numerator, where the difference of squares stands:
The next important skill that may be required when working with rational expressions is isolating the whole part from a fraction.
Let's demonstrate this operation with an example
Let's rewrite the fraction, changing the order of the terms in the numerator:
Both the denominator and the numerator contain the sum x2 + 1. Now you can select the whole part:
You can verify the validity of this transformation by performing it “in the opposite direction”:
Any polynomial can be made into a fraction by assigning it a numerator equal to 1. Let us simplify the formula
Let's replace 2x - 1 with a fraction and subtract:
Simplifying this fraction further is quite difficult, but still possible. To do this, you need to replace the monomial (– 3x2) with the difference (– x2 – 2x2), and 14x with the sum (6x + 8x). Let's see what happens as a result:
You can add more than two fractions. Let us simplify the sum
We will add the terms sequentially, that is, we will first add the first two terms, then add the third, and then the 4th term to the result:
