MHB Partial fraction decomposition (x-3)/(x^2+4x+3)

AI Thread Summary
To perform partial fraction decomposition of (x-3)/(x^2+4x+3), the denominator is factored into (x+3)(x+1). The next step involves expressing the fraction as A/(x+3) + B/(x+1) and eliminating the fractions by multiplying both sides by the denominator. By substituting specific values for x, such as -3 and -1, A and B can be determined easily. The resulting equations yield A = -2 and B = 3, confirming the decomposition. This method effectively simplifies the original expression into its partial fractions.
Guzman10
Messages
2
Reaction score
0
(x-3)/(x^2+4x+3)
After i factor the denominator what do i do next to find A and B?
=(x-3)/(x+3)(x+1)
=A/(x+3)+B/(x+1)
 
Mathematics news on Phys.org
First try to get rid of the fractions. Any ideas?

Then once you do that you can let $x$ equal any value you want. Choose $x$ such that $A$ or $B$ cancels out, then you can solve for the other one. Can you make any progress now? :)
 
Yes, thanks alot
 
Now,to find a,multiplying both sides of the equality by (x+3);

(x-3)/(x+1) =A + B(x+3)/(x+1)

Now,setting x=-3 or x+3=0,makes the expression on the right containing (x+3) vanish. So,

(-3-3)/(-3+1)=A+0 .Then,

A=3

You could find B by a similar method.
 
\frac{x- 3}{x^2+ 4x+ 3}= \frac{x- 3}{(x+ 1)(x+ 3)}= \frac{A}{x+1}+ \frac{B}{x+3}

There are several different ways to do this. The most obvious is to add the fractions on the right side: <div style="text-align: left"><span style="font-family: 'Tahoma'">\frac{x- 3}{(x+ 1)(x+ 3)}</span>&#8203;</div><span style="font-family: 'Tahoma'"><br /> <span style="font-family: 'Tahoma'">=</span></span>\frac{A(x+3)}{(x+ 1)(x+ 3)}+ \frac{B(x+ 1)}{(x+1)(x+ 3)}=\frac{Ax+ 3A+ Bx+ B}{x^2+ 4x^2+ 3}

so we must have (A+ B)x+ (3A+ B)= x- 3. In order that this be true for all x we must have (A+ B)x= x and 3A+ B= -3. Solve the equations A+ B= 1 and 3A+ B= -3 for A and B.

Multiply both sides of the equation by (x+ 1)(x+ 3): x- 3= A(x+ 3)+ B(x+ 1).

And now we can write x- 3= (A+ B)x+ 3A+ B so we must have the A+ B= 1 and 3A+ B= -3 as before. 3A+ B=-3. A+ B= 1" 2A= -4. A= -2. B= 3

Or, since this is to be true for all x we can simply choose two values for x to get to equations. If we take x= 0 (just because it is an easy number) we have -3= 3A+ B again. If we take x= 1 we have -2= 4A+ 2B. Dividing by 2 gives 2A+ B= -1. That last is a new equation but satisfied by the same A and B.

The simplest method is to choose x= -1 and x= -3 because they make the coefficients of one of A and B 0. If x= -1 we have -1- 3= -4= A(-1+ 3)+ B(-1+ 1) so 2A= -4. Taking x= -3 we have -3- 3= -6= A(-3+ 3)+ B(-3+ 1) so -2B= -6.

 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
11
Views
2K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
2
Views
2K
Replies
4
Views
2K
Replies
3
Views
3K
Replies
10
Views
2K
Replies
8
Views
2K
Back
Top