Fractional polynomial addition

In summary, the conversation discusses finding values for A and B in the equation $$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$ by equating the coefficients and constant terms on both sides. This results in two equations which can be solved to find the values of A and B.
  • #1
marksyncm
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5

Homework Statement



Determine whether there exist ##A## and ##B## such that:

$$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$

Homework Equations



None

The Attempt at a Solution


[/B]
First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a remainder, which I interpret as meaning that ##3x^2-5x-2## is the lowest common denominator of ##\frac{A}{3x+1}## and ##\frac{B}{x-2}##. Therefore what I'm looking for is:

$$\frac{A(x-2)}{(3x+1)(x-2)} + \frac{B(3x+1)}{(x-2)(3x+1)}$$

I am unsure as to how to proceed from here. Logically, it seems that we're looking for an ##A## and ##B## such that ##A(x-2) + B(3x+1) = 1##, which results in ##A = \frac{1-B(3x+1)}{x-2}##. However, I'm wondering if this is correct and/or if there's a much more obvious way to find values for A and B?

Thank you.
 
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  • #2
marksyncm said:
it seems that we're looking for an ##A## and ##B## such that ##A(x-2) + B(3x+1) = 1##
That's correct, but note that that equals sign means that the equation must hold for all values of x, which can only happen if the coefficient of x on the LHS is zero. Similarly the constant term (the part that isn't multiplied by x) on the LHS must be 1. Those two requirements give you two equations, which you can solve to find the two unknown parameters A and B.
 
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What is "Fractional Polynomial Addition"?

Fractional polynomial addition is a mathematical operation that involves adding two or more polynomials with fractional coefficients.

How is "Fractional Polynomial Addition" different from regular polynomial addition?

Regular polynomial addition involves adding polynomials with only integer coefficients, while fractional polynomial addition involves adding polynomials with both integer and fractional coefficients.

What are the steps to perform "Fractional Polynomial Addition"?

The steps to perform fractional polynomial addition are:
1. Arrange the polynomials in descending order of degree
2. Group the terms with the same degree
3. Add the coefficients of the grouped terms
4. Write the final answer in standard form with the coefficients in descending order of degree.

Can "Fractional Polynomial Addition" result in a fraction as the final answer?

Yes, since the polynomials being added already have fractional coefficients, the final answer may also have fractional coefficients.

How is "Fractional Polynomial Addition" used in real life situations?

Fractional polynomial addition is used in various fields such as physics, engineering, economics, and statistics. It can be used to model real-life situations and make predictions based on data analysis. For example, in economics, fractional polynomial addition can be used to analyze the relationship between different economic variables and make informed decisions.

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