Now I’m going to teach you how to add and subtract algebraic fractions with the same denominator as well as with different denominators.
The procedure is the same as for adding or subtracting numerical fractions, that is, we need to have the same denominator to add and subtract fractions and when we do not have it, we have to reduce the fractions to a common denominator, with the difference that with algebraic fractions, instead of numbers, we work with polynomials.
Let’s take it one step at a time.
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Algebraic addition and subtraction of algebraic fractions with the same denominator
We start with the addition and subtraction of algebraic fractions that have the same denominator, such as this one:
In this case, the denominator is maintained and the numerators are used. We can leave a single fraction with the common denominator and with the terms of both numerators:
And then group similar terms together in the numerator:
Adding and subtracting algebraic fractions that have the same denominator is that simple. However, great care must be taken in the subtraction of algebraic fractions, since the minus sign affects all the terms of the numerator of the fraction behind it.
Let’s look at an example with subtraction of algebraic fractions to help you understand it better:
We have the same denominator and therefore, we can join all the numerators into one. But now, in front of the last fraction we have a minus sign and as I was telling you before, it affects the two terms of the numerator of the fraction behind it. Therefore, for this to continue to be the case, the terms affected by the minus sign must be enclosed in parentheses:
In the next step, we remove the parenthesis, changing the sign to the terms inside it:
And finally, we group similar terms in the numerator:
A very common mistake is not to bracket the terms of the numerator of the fraction that is preceded by a minus sign. If we don’t, we only change the sign in the first place, which is a mistake:
Algebraic addition and subtraction of algebraic fractions with different denominator
Now I will explain how to add and subtract algebraic fractions with different denominators.
Keep in mind that you can only add or subtract algebraic fractions that have the same denominator, so if they have different denominators, you must first reduce them to common denominator, such and so to add and subtract numerical fractions with different denominators.
We obtain the common denominator by calculating the lowest common multiple of the denominators.
For example, we have additions and subtractions of algebraic fractions, whose denominators are not the same:
The first step will be to reduce them to common denominator by calculating the lowest common multiple of the denominators:
To do this, the polynomials must first be broken down. The polynomials of the second and third algebraic fraction cannot be broken down because they are of degree 1:
Therefore, we have the factors (x+1) and (x-1), without any exponent (or elevated to 1), so the lowest common multiple will be the multiplication of both:
We are going to transform every algebraic fraction we already had into an equivalent algebraic fraction with the common denominator we just calculated.
We put our new denominator in each of the fractions, ready to start calculating the new numerator for each fraction:
To transform one algebraic fraction into another that is equivalent, we have to multiply the numerator and denominator by the same polynomial.
By placing the common denominator directly, we have already multiplied the original denominator by a polynomial, so we need to multiply the numerator by the same polynomial so that the algebraic fraction is equivalent.
How do we obtain that polynomial by which each numerator must be multiplied?
It is obtained by dividing the common denominator by the original denominator of each algebraic fraction:
Let’s look at it with the first algebraic fraction of the operation in the example:
Initially we had in the denominator the polynomial (x²-1) and in the new equivalent fraction we have as denominator (x+1):
In this case, the common denominator and the original denominator coincide because (x²-1)=(x+1)(x-1), therefore, having the same denominator, the numerator is multiplied by 1, or in other words, it remains the same:
If we calculate the polynomial by which to multiply the numerator using the previous formula, we obtain that 1:
Let’s see which is the polynomial by which we have to multiply the second fraction:
We had the original denominator (x+1) and now we have (x+1) (x-1) which is the common denominator. By dividing the common denominator by the original denominator we are left with (x-1):
That is the polynomial by which we have to multiply the numerator so that the algebraic fraction is equivalent to the original one:
And finally, we do the same thing with the third algebraic fraction:
We divide the common denominator into the original denominator:
And we multiply the result by the numerator:
In this way we have obtained the three equivalent algebraic fractions with the common denominator:
They already have the same denominator and can therefore be added and subtracted, but first, we multiply in the numerators:
Now we can unite all the numerators in a single fraction with the common denominator. Keep in mind, as always, that the minor sign affects all the terms of the numerator behind it and that’s why we must enclose those terms in parentheses (I won’t get tired of repeating it):
We remove the parentheses of the numerator:
We group terms in the numerator and in the last step, we multiply in the denominator to obtain the final result:
Let’s take another example step by step to make it even clearer:
We have an addition and subtraction of algebraic fractions with different denominators, so we have to reduce them to a common denominator. For this we obtain the lowest common multiple of the denominators:
The denominators of the first and second algebraic fraction cannot be broken down. We decompose the denominator of the third algebraic fraction:
We have as factors x, x² and (x+1). From x to x² we get x² because we have a bigger exponent and we also get (x+1), so the mcm is:
We leave each algebraic fraction ready with the new denominator:
We multiply each numerator by the polynomial obtained by dividing the common denominator by the original denominator:
We put all the numerators together in a single fraction, taking the minus sign very carefully into account:
We operate on the numerator by removing parentheses:
And finally we group terms and operate in the denominator to get the final result:
Solve the following additions and subtractions of algebraic fractions: