In simple terms, fractions mean a part of the whole. For example, 4/5, 2/3 are fractions and can also be called common fractions. The upper number or the dividend is known as the numerator, while the lower number or the divisor is known as the denominator. Let us discuss below what is a partial fraction, and what is the process of finding them.
As we know, fractions can be classified as a proper and improper fraction; in the same way, even partial fractions can also be classified as proper and improper, as will be seen later. In arithmetic terms, a fraction signifies the division of integers. In elementary algebra, we studied how different fractions can be added up by finding their LCM and then adding all the fractions.
Like if we have to add the fractions: 3/(y-1) and 1/(y+2)
We will get (4y+5)/[(y-1)(y+2)]
Let us see the reverse process of the above, which means that we will see how to split a fraction into several fractions, the denominators of whom are the factors of the denominator of that fraction. These are known as partial fractions.
When a single rational fraction is expressed as the sum of two or more single rational fractions, it is called partial fraction resolution.
Rational numbers, as we know, are numbers expressed as x/y, where y cannot be zero in the same ways rational fractions are those where the quotient of two polynomials A(x)/B(x) where B(x) is not equal to zero, without any common factors. Rational fractions are of two types: proper and improper.
The proper rational fraction is one where the degree of the numerator is less than the degree of the denominator.
The improper rational fraction is one where the degree of the numerator is greater than or equal to the degree of the denominator. An improper fraction can be written as the sum of a polynomial and a proper fraction.
A proper fraction can be converted to partial fractions as:
If in the denominator (ax+b) linear factor is there, and it is non-repeating, the partial fraction of this will be of the form A/(ax+b), where A stands for the constant whose value needs to be found out.
If in the denominator ax+b occurs ‘n’ number of times like (ax+b) to the power n, then there will be ‘n’ partial fractions like A1/(ax+b) + A2/(ax+b)2……..
If there is a quadratic factor in the denominator ax2+bx+c, and is non-repeating, the partial fraction will then be of the form of (ax+b)/(ax2+bx+c) where a and b are constants to be found out.
If the quadratic factor occurs ‘n’ times in the denominator, then there will be ‘n’ number of partial fractions.
The evaluation of the coefficients in the case of partial fractions is done based upon the following theorem, that if the polynomials stand equal for all of the values of the variables, then the coefficients which have the same degree on both of the sides are equal.
Fractions are an important concept in mathematics, and hence the students should make sure to have enough practice on the same. Many websites these days offer a lot of study resources online, one such useful resource being the worksheets. Math worksheets are a wonderful way to improve mathematical concepts through practice. Cuemath offers a variety of math worksheets on different topics. These are easy and free to download, and students can print them easily. Solving them will not only increase the logical thinking and analytical skills but will also give them the confidence to solve different kinds of math problems through their strategic approach.
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