NationMaster - Encyclopedia: Partial fractions

In algebra, the partial fraction decomposition or (partial fraction expansion) of a rational function expresses the function as a sum of fractions, where: Linear algebra lecture at Helsinki University of Technology This article is about the branch of mathematics; for other uses of the term see algebra (disambiguation). ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

the denominator of each term is a power of an irreducible (not factorable) polynomial and
the numerator is a polynomial of smaller degree than the denominator.

See partial fractions in integration for an account of their use in finding antiderivatives. They are also used in calculating the inverse of transforms; such as the Laplace transform, or the Z-transform. In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In integral calculus, the use of partial fractions is required to integrate the general rational function. ... In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

Just which polynomials are irreducible depends on which field of scalars one adopts. Thus if one allows only real numbers, then irreducible polynomials are of degree either 1 or 2. If complex numbers are allowed, only 1st-degree polynomials can be irreducible. If one allows only rational numbers, then some higher-degree polynomials are irreducible. This article presents the essential definitions. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

1 Some examples
2 Basic principles
3 Examples
4 See also

Some examples

Distinct first-degree factors in the denominator

Suppose it is desired to decompose the rational function

${x+3 over x^2-3x-40},$

into partial fractions. The denominator factors as

$(x-8)(x+5),$

and so we seek scalars A and B such that

${x+3 over x^2-3x-40}={x+3 over (x-8)(x+5)}={A over x-8}+{B over x+5}.$

One way of finding A and B begins by "clearing fractions", i.e., multiplying both sides by the common denominator (x − 8)(x + 5). This yields In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ...

$x+3=A(x+5)+B(x-8).,$

Collecting like terms gives

$x+3=(A+B)x+(5A-8B).,$

Equating coefficients of like terms then yields:

$begin{matrix} A & + & B & = & 1 5A & - & 8B & = & 3 end{matrix}$

The solution is A = 11/13, B = 2/13. Thus we have the partial fraction decomposition

An irreducible quadratic factor in the denominator

In order to decompose

${10x^2+12x+20 over x^3-8}$

into partial fractions, first observe that

$x^3-8=(x-2)(x^2+2x+4).,$

The fact that x² + 2x + 4 cannot be factored using real numbers can be seen by observing that the discriminant 2² − 4(1)(4) is negative. Thus we seek scalars A, B, C such that In mathematics, a discriminant is an expression which discriminates qualities of algebraic structures. ...

${10x^2+12x+20 over x^3-8}={10x^2+12x+20 over (x-2)(x^2+2x+4)}={A over x-2}+{Bx+C over x^2+2x+4}.$

When we clear fractions, we get

$10x^2+12x+20=A(x^2+2x+4)+(Bx+C)(x-2).,$

We could proceed as in the previous example, getting three linear equations in three variables A, B, and C. However, since solving such systems becomes onerous as the number of variables grows, we try a different method. Substitution of 2 for x in the identity above makes the entire second term vanish, and we get

$10cdot 2^2+12cdot 2+20=A(2^2+2cdot 2+4),,$

i.e., 84 = 12A, so A = 7, and we have

$10x^2+12x+20=7(x^2+2x+4)+(Bx+C)(x-2).,$

Next, substitution of 0 for x yields

$20=7(4)+C(-2),,$

and so C = 4. We now have

$10x^2+12x+20=7(x^2+2x+4)+(Bx+4)(x-2).,$

Substitution of 1 for x yields

$10+12+20=7(1+2+4)+(B+4)(1-2),,$

and so B = 3. Our partial fraction decomposition is therefore:

${10x^2+12x+20 over x^3-8}={7 over x-2}+{3x+4 over x^2+2x+4}.$

A repeated first-degree factor in the denominator

Consider the rational function

${10x^2-63x+29 over x^3-11x^2+40x-48}.$

The denominator factors thus:

$x^3-11x^2+40x-48=(x-3)(x-4)^2.,$

The multiplicity of the first-degree factor (x − 4) is more than 1. In such cases, the partial fraction decomposition takes the following form:

${10x^2-63x+29 over x^3-11x^2+40x-48}={10x^2-63x+29 over (x-3)(x-4)^2}={A over x-3}+{B over x-4}+{C over (x-4)^2}.$

Repeated factors in the denominator generally

For rational functions of the form

${bullet over (x+2)(x+3)^5}$

(where the " $bullet$ " may be any polynomial of sufficiently small degree) the partial fraction decomposition looks like this:

${A over x+2}+{B over x+3}+{C over (x+3)^2}+{D over (x+3)^3}+{E over (x+3)^4}+{F over (x+3)^5}.$

The general pattern may be quickly guessed.

For rational functions of the form

${bullet over (x+2)(x^2+1)^5}$

with the irreducible quadratic factor x² + 1 in the denominator (where again, the " $bullet$ " may be any polynomial of sufficiently small degree), the partial fraction decomposition looks like this:

${A over x+2}+{Bx+C over x^2+1}+{Dx+E over (x^2+1)^2}+{Fx+G over (x^2+1)^3}+{Hx+I over (x^2+1)^4}+{Jx+K over (x^2+1)^5},$

and a similar pattern holds for any other irreducible quadratic factor.

[Still to be added: high-degree polynomials in the numerator.]

Use in deriving the logistic general equation

In many beginning calculus courses, partial fractions are introduced as a way to derive the general equation for a logistic function. Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. ...

Logistic functions model a population which grows until it reaches a limit. The rate of change for the function is proportional (constant k) to both the population reached (P) and the fraction of the total carrying capacity (M) remaning. Thus: As population density increases, birth rates decrease and death rates increase. ...

$frac{dP}{dt}=kPleft({M-P over M}right)$

$int {1 over P(M-P)}, dP = int {k over M}, dt$

${1 over P(M-P)} = {A over P} + {B over (M-P)}$

1 = A (M - P) + B P

1 = A M - A P + B P

$A = B, A= {1 over M}, B = {1 over M}$

$int {{1 over MP} + {1 over M(M-P)}},dP = int {k over M}, dt$

$int {{1 over P} + {1 over (M-P)}},dP = int {k}, dt$

$ln{P over (P-M)} = kt + C$

${P over P-M} = e^{kt + C}$

${M-P over P} = e^{-kt - C}$

${M over P} - 1 = e^{-kt - C}$

$P = {M over {e^{-kt - C} + 1}}$

$P = {M over {Ae^{-kt} + 1}}$

Basic principles

The basic principles involved are quite simple; it is the algorithmic aspects that require attention in particular cases.

Assume a rational function R(x) in one indeterminate x has denominator that factors as See: indeterminate (variable) statically indeterminate Division by zero This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

P(x)Q(x)

over a field K (we can take this to be real numbers, or complex numbers). If P and Q have no common factor, then R may be written as This article presents the essential definitions. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ...

A/P + B/Q

for some polynomials A(x) and B(x) over K. The existence of such a decomposition is a consequence of the fact that the polynomial ring over K is a principal ideal domain, so that In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...

CP + DQ = 1

for some polynomials C(x) and D(x) (see Bézout's identity). In number theory, BÃ©zouts identity, named after Ã‰tienne BÃ©zout, is a linear diophantine equation. ...

Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomials. To take this further, if required, write In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ...

G(x)/F(x)ⁿ

as a sum with denominators powers of F and numerators of degree less than F, plus a possible extra polynomial. This can be done by the Euclidean algorithm, polynomial case. In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ...

Therefore when K is the complex numbers and we can assume F has degree 1 (by the fundamental theorem of algebra) the numerators will be constant. When K is the real numbers we can have the case of In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree â‰¥ has some complex root. ...

degree F = 2,

and a quotient of a linear polynomial by a power of a quadratic will occur. This therefore is a case that requires discussion, in the systematic theory of integration (for example in computer algebra). A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...

Examples

As an introductory example we take the rational function

$frac{x}{x^2-1}.$

This by the difference of two squares identity can also be written as

$frac{x}{(x+1)(x-1)},$

which can be transformed further. Consider an identity

$frac{A}{x+1}+frac{B}{x-1}=frac{x}{x^2-1},$

where A and B are constants. In more explicit form,

$!, A(x-1)+B(x+1)=x.$

We know that the constants on one side of an expression must equal those on the other side. On the left hand side, the constants are −A and B, and on the right, the constant is simply 0. So, comparing constants on both sides of the expression, we can see that

$B-A=0,,$

i.e.

A = B .

Now, in the same way, we know that the number of x terms on the left must equal the number of x's on the right. Therefore, looking at x terms on both sides,

$Ax+Bx=x,,$

therefore

$A+B=1,,$

and so, given that

A = B

, we can say that

$A + A$	$= 1$
$2 A$	$= 1$
$A$	$= B = 0.5$

Finally we find:

$frac{x}{x^2-1}=frac{1}{2}cdotfrac{1}{x+1}+frac{1}{2}cdotfrac{1}{x-1}$

$frac{x}{x^2-1}=frac{1}{2(x+1)}+frac{1}{2(x-1)}$

which holds true for all x ≠ ±1.

The preceding example can be generalized to the following situation:

Assume that Q(x) is a monic polynomial of some degree n which over the underlying field K decomposes into linear factors

$Q(x)=prod_{i=1}^n (x-x_i),$

where all

x i

are pairwise different. In other words Q has simple roots (over K). If P(x) is any polynomial of degree $le n-1$ then according to the Lagrange interpolation formula (see Lagrange form) P(x) can be uniquely written as a sum (the Lagrange form representation)

$P(x)=sum_{j=1}^n P(x_j)L_j(x;x_j),$

where $, L_j(x;x_j)$ is the Lagrange polynomial

$L_j(x;x_j)=prod_{kle n,, kne j} {{(x-x_k)}over {(x_j-x_k)}}.$

Dividing the Lagrange representation on the right side termwise by the polynomial Q(x) in its factored form one obtains

${P(x)over Q(x)} =sum_{j=1}^n {P(x_j)over {prod_{k le n, , kne j} (x_j-x_k)}} ,cdot {1 over {x-x_j}}.$

This is the partial fraction decomposition

${P(x)over Q(x)} =sum_{j=1}^n c_j cdot {1 over {x-x_j}}$

of the rational function $, R(x)=P(x)/Q(x)$ with coefficients

The first example can be obtained as the special case .

In mathematics, the difference of two squares refers to the identity a2 âˆ’ b2 = (a + b)(a âˆ’ b) from elementary algebra. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... This article is about the term degree as used in mathematics. ... This article presents the essential definitions. ... In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. ...

Contents

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