Case 4. Repeated quadratic term in denominator

[latex s=2]F(x) = \frac{2x^3+5x}{(x^2+3)^2}[/latex] [latex s=3]F(x) = \frac{2x^3+5x}{(x^2+3)^2} = \frac{Ax+B}{(x^2+3)}+\frac{Cx+D}{(x^2+3)^2}[/latex] Multiply an divide [latex](x^2+3)^2[/latex] with [latex]Ax+B [/latex] [latex s=3]\frac{2x^3+5x}{(x^2+3)^2} = \frac{(Ax+B)(x^2+3)}{(x^2+3)(x^2+3)}+ \frac{Cx+D}{(x^2+3)^2}[/latex] [latex s=3]\frac{2x^3+5x}{(x^2+3)^2} = \frac{(Ax+B)(x^2+3)}{(x^2+3)^2}+ \frac{Cx+D}{(x^2+3)^2}[/latex] [latex s=3]\frac{2x^3+5x}{(x^2+3)^2} = \frac{(Ax+B)(x^2+3)+Cx+D}{(x^2+3)^2}[/latex] [latex s=3]\frac{2x^3+5x}{(x^2+3)^2} = \frac{Ax^3+Bx^2+3Ax+3B+Cx+D}{(x^2+3)^2}[/latex] [latex s=3]\frac{2x^3+5x}{(x^2+3)^2} = \frac{Ax^3+Bx^2+3(A+C)x+(3B+D)}{(x^2+3)^2}[/latex] Comparing the coefficient we get [latex]\framebox[1.1\width]{A=2}[/latex] [latex]\framebox[1.1\width]{B=0}[/latex] [latex]3A+C=5 \longmapsto \framebox[1.1\width]{C=-1}[/latex] [latex]3B+D=0 \longmapsto \framebox[1.1\width]{D=0}[/latex] Put the value […]

Case 3. Quadratic term in denominator.

[latex s=2]F(x) = \frac{8x^2-12}{x(x^2+2x-6)}[/latex] [latex s=2]F(x) = \frac{8x^2-12}{x(x^2+2x-6)} = \frac{A}{(x)}+\frac{Bx+C}{(x^2+2x-6)}[/latex] Multiply by  [latex s=1]x(x^2+2x-6) [/latex] with above equation on both sides we get [latex s=1]\Big(x(x^2+2x-6)\Big) \Big(\frac{8x^2-12}{x(x^2+2x-6)}\Big) = \Big(x(x^2+2x-6)\Big) \Big(\frac{A}{(x)}\Big)+ \Big(x(x^2+2x-6)\Big) \Big(\frac{B}{x(x^2+2x-6)}\Big)[/latex] [latex]8x^2-12 = A(x^2+2x-6)+x(Bx+C)[/latex] We could choose [latex]x=0[/latex] to get the value of A, but that’s the only constant that we could get using this method […]

Case 2. Repeated linear factor in denominator.

[latex s=2]F(x) = \frac{4x^2}{(x-1)(x-2)^2}[/latex] [latex s=2]F(x) = \frac{4x^2}{(x-1)(x-2)^2} = \frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-2)^2}[/latex] Multiply by [latex](x-1)(x-2)^2[/latex] with above equation on both sides we get [latex s=1]\Big((x-1)(x-2)^2\Big) \Big(\frac{4x^2}{(x-1)(x-2)^2}\Big) = \Big((x-1)(x-2)^2\Big) \Big(\frac{A}{(x-1)}\Big)+ \Big((x-1)(x-2)^2\Big) \Big(\frac{B}{(x-2)}\Big)+ \Big((x-1)(x-2)^2\Big) \Big(\frac{C}{(x-2)^2}\Big)[/latex] [latex s=1]4x^2 = A(x-2)^2+B(x-1)(x-2)+C(x-1)[/latex]  …………(1) Now we have [latex] (x-1)[/latex] and [latex](x-2)^2[/latex] [latex]x-1=0 \longmapsto x=1[/latex] [latex](x-2)^2=0 \longmapsto x=2[/latex] Put [latex]\textbf{x=1}[/latex] and [latex]\textbf{x=2}[/latex] in equation (1) […]

Case 1. Distinct linear factor in denominator

Let us see the how to slove when distinct linear factor is in denominator. Distinct Linear Factor: [latex s= 2]F(x) = \frac{8x-42}{x^2+3x-18}[/latex] [latex s=2]F(x) = \frac{8x-42}{x^2+3x-18} = \frac{8x-42}{(x+6)(x-3)} = \frac{A}{x+6}+\frac{B}{x-3}[/latex] Multiply by [latex]\textbf(x+6)(x-3)[/latex] on both sides with above equation we get [latex s=-1]\Big((x+6)(x-3)\Big)\Big(\frac{8x-42}{x^2+3x-18}\Big) = \Big(\Big((x+6)(x-3)\Big)\Big(\frac{A}{x+6}\Big)\Big) + \Big(\Big((x+6)(x-3)\Big)\Big(\frac{B}{x-3}\Big)\Big)[/latex]   [latex]8x-42=A(x-3)+B(x+6)[/latex]  …………(1) Now we have [latex]\textbf{(x+6)}[/latex] and […]

Partial Fraction

Partial Fraction Method

First Method The method for computing partial fraction decomposition applies to all rational functions with one qualification: ‘The degree of the numerator must be less than the degree of the denominator’ Works for Proper Rational Expressions, where the degree of the numerator is less than the bottom. Note: When order/degree of numerator is not less […]

MATLAB code for Bisection method

How to run the code: 1. Download the m file and open it in MATLAB. 2. Now there are 6 input arguments and minimum 3 argument is required to run the program. 3. First one is function. You can write the function as below func=inline(‘-0.5*x^2+2.5*x+4.5’); or [email protected](x) -0.5*x^2+2.5*x+4.5; 4. Third and fourth xl and xu which […]

Bisection method feature image

Bisection method with example.

The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a sub-interval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to […]

spectrum

What is Spectrum ?

Definition: A visual display dispersion of components of white light when it is passed through a prism is called spectrum. The study of spectrum is called spectroscopy. The instrument used to analyze spectrum is called spectrometer. Explanation: Ordinary white light consists of radiations of all wavelengths. After passing through the prism, white light is splitted […]

Photocopier

Inkjet printers

An inkjet printer is a type of printer which uses electric charge in its operation. While shuttling back and forth across the paper, the inkjet printer “ejects” a thin stream of ink. The ink is forced out of a small nozzle and breaks up into extremely small droplets. During their flight, the droplets pass through […]

Plastic raw materials

What is plastic ?

Definition: Plastics are macromolecules, which are formed by the polymerization of simple molecules. Explanation: In other words we can say that plastic are the polymers formed from monomers. The phenomenon in which simple and smaller molecules are combined together to form complex and large molecules, is called polymerization. The simple and smaller molecules are called […]