algebraic fractions they are expressions that have at least one unknown in the denominator. Unknowns are unknown numbers usually represented by letters. In this way, it is possible to define the basic mathematical operations also for the algebraic fractions.
The technique used to add and subtract algebraic fractions is exactly the same used for numerical fractions, including divided into two cases. The difference is in the mathematical devices used to enable calculations, such as polynomial factorization or potency properties.
Case 1: Algebraic fractions with equal denominators
when the algebraic fractions have the same denominators, they can be added or subtracted directly, just repeating the common denominator and performing the operation only with the numerators. Note the following example:
16xk2 – 10xk2 = 16xk2 – 10xk2 = 6xk2
y y y y
Regardless of the form the algebraic fractions or if the numerators are similar terms, just keep the denominator and operate the numerators with the rules of plus signs.
Case 2: Algebraic fractions with different denominators
when the algebraic fractions to be added or subtracted have different denominators, it is necessary to find equivalent fractions to them that have the same denominators for later add them up. The procedure for finding these fractions is the same as for adding numeric fractions: calculate the least common multiple of the denominators, find the equivalent fractions and then perform the addition/subtraction of fractions with equal denominators. Note the following addition example:
a + b + 4th2 – a - b
tab2 - B2 a + b
Minimum common multiple of denominators
Calculating the MMC of whole numbers is not a challenging task. However, the minimum between polynomials takes a lot of practice. To learn how to perform this calculation, read the article “Least Common Multiple of Polynomials” on here.
In short, it is necessary to factor the polynomials of the denominators and then multiply all the factors that have the same base with a higher exponent without repetitions.
Therefore, the denominators in the example above are: a – b, (a – b)(a + b), which is the factored form of a2 - B2, and a + b. The MMC between these denominators is (a – b)(a + b), which is precisely the product of factors of the same base with the highest exponent without repetitions. Once this is done, rewrite the fractions in the example using the new common denominator and leaving spaces to find the equivalent numerators.
a + b + 4th2 – a - b = + –
tab2 - B2 a + b (a - b) (a + b) (a - b) (a + b) (a - b) (a + b)
Find the equivalent fractions
To find the numerator of the first fraction equivalent, divide the MMC found by the denominator of the first given fraction and then multiply the result by its numerator. The result of this will be the numerator of the first fraction equivalent. For the others, repeat the process using the respective fractions.
Thus, the numerator of the first fraction equivalent is the result of (a – b)(a + b) divided by a – b and multiplied by a + b. This results in (a + b)2. Continuing the calculations for the others fractions and putting the results in their respective numerators, we have:
a + b + 4th2 – a - b = (a + b)2 + 4th2 – (a - b)2
tab2 - B2 a + b (a - b) (a + b) (a - b) (a + b) (a - b) (a + b)
Perform addition/subtraction
In this last step, the proposed operations are carried out effectively. Watch:
(a + b)2 + 4th2 – (a - b)2 =
(a – b)(a + b) (a – b)(a + b) (a – b)(a + b)
(a + b)2 + 4th2 – (a – b)2 =
(a - b) (a + b)
The2 + 2ab + b2 + 4th2 - a2 + 2ab - b2 =
(a - b) (a + b)
2b + 4a2 + 2b =
(a - b) (a + b)
4th2 + 4ab =
(a - b) (a + b)
It is also in this step that the result is simplified through factorization of polynomials and sometimes properties of powers.
4th2 + 4ab =
(a - b) (a + b)
4a (a + b) =
(a - b) (a + b)
4The
a - b
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/adicao-subtracao-fracoes-algebricas.htm
