More polynomial division

In summary, the conversation discusses a problem involving polynomial division and finding the integral of a function. The person asking for help is struggling to find the correct equations to solve for the variables A, B, and C. There are two suggested methods for solving the problem, setting the coefficients of the same powers equal and taking specific values for x.
  • #1

ssb

119
0
Some of you may have read my troubles that I had in this thread
https://www.physicsforums.com/showthread.php?t=162079

Now I am having more troubles with more advanced problems.

I started with this problem:

[4y^2 - 7y - 12] / [(y)(y+2)(y-3)]

The problem is that I set it up the same way as I did in that thread and I am able to solve for A, B, and C:
A= 3
B= 9/5
C= 1/5

so I integrate 3/y + (9/5)/(y+2) + (1/5)/(y-3)

and i get

3ln(y) + 9/5ln(y+2) + 1/5ln(y-3)

Beautiful right? Well I am suppost to find the integral over the area from 1 to 2.

When I plug in 1 and 2 into the 1/5 ln (y-3) it yields a negetive number and you can't take the ln of a negetive number!

Im sure I am making the mistake in the polynomial division somewhere but i don't know. I am thinking there is a way to simplify the original question first before i solve for A B and C... right? any insite would be wonderful!
 
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  • #2
Ok so I forgot to mention that I also tried to take the absolute values of the value inside the ln and it still did not work.
 
  • #3
Oh ok. You didn't set up your equations right. With a denominator with three factors, it becomes (you can see this by yourself):

[tex] A(y)(y+2) + B(y)(y-3) + C(y+2)(y-3) = 4y^2 - 7y - 12 [/tex]

[tex] A(y^{2} + 2y) + B(y^{2} - 3y) + C(y^{2} - y - 6) = 4y^2 - 7y - 12 [/tex]

[tex] (A + B + C)y^{2} + (2A - 3B - C)y - 6C = 4y^2 - 7y - 12 [/tex]If we want A, B and C to be constants, we compare each term with one another (equality of polynomials theorem). We find C = 2, and then we get [tex]2A - 3B - 2 = -7 [/tex] and [tex] A + B + 2 = 4 [/tex]. Now we have two unknowns with two equations and we solve.
 
Last edited:
  • #4
What Werg22 is talking about is setting the coefficients of the same powers equal.


Another very nice method is, since the equation must be true for all x, to take specific values for x. Since the left hand side of the equation Werg22 gave you involve factors y, y+2, and y- 3, let y= 0, -2, and 3 in succesion. Your equations become:
x= 0 0+ 0+ C(2)(-3)= -12 so C= 2
x=-2 0+ B(-2)(-5)+ 0= 4(-2)2- 7(-2)- 12= 16+14-12= 18 so B= 9/5
x= 3 A(3)(5)+ 0+ 0= 4(3)2-7(3)-12= 36- 21-12= 3 so A= 1/5.
 

1. What is the purpose of polynomial division?

Polynomial division is a mathematical process used to divide one polynomial expression by another. It is often used to simplify or solve equations involving polynomials, and to find the remainder or quotient of a polynomial division.

2. How do you perform polynomial division?

To perform polynomial division, you can use the long division method or synthetic division method. In long division, the dividend (the polynomial being divided) is divided by the divisor (the polynomial dividing the dividend) in a step-by-step process. In synthetic division, the coefficients of the dividend are used to simplify the division process.

3. Can polynomial division be used to solve real-world problems?

Yes, polynomial division can be used to solve real-world problems in fields such as engineering, physics, and economics. It can be used to model and solve equations involving polynomial functions, which are often used to describe real-world situations.

4. What is the relationship between polynomial division and factoring?

Polynomial division and factoring are closely related concepts. The remainder obtained from polynomial division is equal to the remainder obtained from factoring the dividend and divisor into their individual terms. This means that polynomial division can be used to check the accuracy of factoring and vice versa.

5. Are there any special cases in polynomial division?

Yes, there are a few special cases in polynomial division. These include division by a monomial (a polynomial with one term) and division by a binomial (a polynomial with two terms). In these cases, the division can be simplified using specific rules and formulas.

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