Theory of Partial Fractions

Partial Fractions:

Addition of Rational Expressions:

In arithmetic, we learned how to add up the fractions. We found the least common denominator and then multiplied both the numerator and denominator of each and every term by what was required to complete the common denominator.

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In algebra, we have carried that procedure on to addition of rational expressions. We once again multiplied the numerator and denominator of each and every term by what was missing from the denominator of the term.

With Partial Fraction Decomposition, we are going to reverse the procedure and decompose a rational expression to two or more simpler proper rational expressions which were added together.

Partial Fraction Decomposition:

The Partial Fraction Decomposition just works for proper rational expressions, that is, the degree of numerator should be less than the degree of denominator. When it is not, then you should execute long division first, and then execute the partial fraction decomposition on the rational part (that is, the remainder over the divisor). After you have done the partial fraction decomposition, simply add back in the quotient portion from the long division.

We have learned that each and every polynomial with real coefficients can be factored by using only linear and irreducible quadratic factors. This signifies that there are only two kinds of factors which we have to worry about.

Linear Factors:

When the partial fractions we are decomposing the rational expression must be appropriate, then the only thing which can be over a linear factor is a constant. Therefore, for each and every linear factor in the denominator, we will need a constant in over that in numerator.

Irreducible Quadratic factors:

When the partial fractions we are decomposing the rational expression must be appropriate, then an irreducible quadratic factor could encompass a linear term and/or a constant term in the numerator. Therefore, for each and every irreducible quadratic factor in the denominator, we will require a linear term and a constant term in numerator.

Repeated Factors:

Let consider a fraction in which the denominator is 8. Does that signify that the denominator of each term being added altogether had to be an 8? No, the denominators could have been 2, 4 or 8 as the common denominator among 2, 4 and 8 is 8. Implications of this for partial fraction decomposition are that whenever you have a repeated factor (that is, a factor with a multiplicity other than one); you need to comprise a factor in the expansion for each and every power possible.

For illustration, if you encompass an (x-2)3, you will need to comprise an (x-2), an (x-2)2, and an (x-2)3.

The exponents of 2 or 3 do not change whether the factor is quadratic or linear, only how many times the factor is there. Each of such (x-2) factors would receive a constant term in numerator as x-2 is linear, no matter what power it is elevated to.

Improper Fractions:

The proper fraction is a fraction where the numerator is less than the denominator. For rational expressions, that signifies the degree in numerator is less than the degree in denominator. When you contain an improper fraction, you first have to execute long division to obtain a quotient and a remainder.

Leave the quotient part only; however execute the partial fraction decomposition on the fraction which remained.

The Basic Equation:

After setting the partial fraction equation, we multiply both sides of the equation by the least common denominator to get rid of fractions. The resultant equation, devoid of the fractions, is termed as the Basic Equation.

Technique 1: Pick Values for x

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Whenever there are linear factors, the simplest way to do the decomposition is most likely to pick the nice values for x.

In illustration to the right, we take each and every factor in the denominator and give it its own term on right side. As each factor in the denominator is linear and the rational expressions require being appropriate, a constant term was positioned in each numerator of each term. We multiply via by the least common denominator to arrive at the equation devoid of the fractions. This equation is termed as the Basic Equation and is:

x + 2 = A (x - 4) + B x

‘Nice’ values for x are such that which cause each linear factor to be zero. In this case, x = 4 and x = 0 are nice. ‘Nice’ values will cause each and every term apart from for one to drop out of the equation, and therefore you will be capable to find out the value of a variable very fast.

When you suppose x = 0, then the B term drops out and you get:

0 + 2 = A (0 - 4) or 2 = -4A. Solving that provides A = -1/2.

Whenever you suppose x = 4, then the A term drops out and you get:

4 + 2 = B (4) or 6 = 4B. Solving that provides B = 3/2.

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After you have determined the values of each and every constant, it is significant that you plug such values back to the decomposition. Do not just stop with A = -1/2, B = 3/2 as someone else might have defined A and B differently. The accurate answer is to place them back to the decomposition and simplify when essential. The simplification would be to decrease the number of signs (do not contain the first term negative when possible) and to remove the compound fractions by factoring out the least common denominator of numerators.

Note: There are just as many nice values as there are distinct (various) linear factors. When there are repeated linear factors or irreducible quadratic factors (that is, repeated or not), you will not have sufficient ‘nice’ values to pick. In cases similar to that, you will have to pick convenient however not-so-nice values and then replace the known constants to the equation to determine the other constants. Plug in simple numbers such as x = 0, x = 1 and so on.

You will require picking as many values for x as there are constants to be determined.

Technique 2: Creating a System of Linear Equations

The first method of picking the values for x works very well whenever all the factors are distinct linear factors. When there are any linear factors, then first technique is still probably the best method to use. Though, if there are just irreducible quadratic factors, then the technique of picking values for x can become confused.

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There is another way to do such problems (actually, this method will work when there are linear factors, just that the other is simpler and faster).

The very first part of the procedure is similar. Go ahead and write out the decomposition, comprising constant terms over linear factors and linear and constant terms over irreducible quadratic factors. Then, multiply via by the least common denominator to determine the fundamental equation. That much is similar. The basic equation is:

2x2 + x + 8 = Ax (x2 + 4) + B (x2+4) + Cx + D

Now the difference comes.

Go further on and expand (multiply out) the basic equation:

2x2 + x + 8 = Ax3 + 4Ax + Bx2+ 4B + Cx + D

And re-group the terms by common powers of variable x.

2x2 + x + 8 = Ax3 + Bx2 + 4Ax + Cx + 4B + D

Now factor out the x's by power;

2x2 + x + 8 = (A) x3 + (B) x2 + (4A + C) x + (4B + D) (1)

The next portion works as if two polynomials are going to be equivalent, they should have similar number of similar terms on both the sides. Therefore the trick here is to equate both the sides of equation altogether by equating similar terms.

Set x3 terms on the left (of which there are none) equivalent to x3 terms on the right (of which there are A) to arrive at first equation 0 = A. Well, that was quite simple, and you already have the value for A.

Set x2 terms on the left (of which there are 2) equivalent to the x2 terms on the right (of which there are B) to arrive at second equation of 2 = B. Now you know B.

For x's, there is 1 on the left and 4A + C on the right, therefore 1 = 4A+C.

For constants, there are 8 on the left and 4B+D on the right, therefore 8 = 4B+D.

You might have to resolve a system of linear equations; however that is why this a part on partial fractions.

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After some solving, we arrive at A = 0, B = 2, C = 1, and D = 0 and we are ready to plug such values back to the original decomposition to arrive at final answer.

Be sure and simplify the answer whenever essential.

This illustration was really pretty simple as you were able to determine A and B right off the bat. Most of the times, you will require to solve a much bigger system of equations to determine the values.

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