DW123's question at Yahoo Answers regarding partial fraction decomposition

In summary, the question is asking how to decompose the equation $f(x)=(-10x-7)/(2x^2-17x+21)$ into two simpler fractions. The solution involves factoring the numerator and denominator, assuming a decomposition form, and using the Heaviside cover-up method to find the values of the variables. The final answer is $f(x)=4/(2x-3)-7/(x-7)$. The poster also invites others to post related questions on a math help forum.
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MarkFL
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Hello DW123,

We are give to decompose:

$\displaystyle f(x)=\frac{-10x-7}{2x^2-17x+21}$

Our first step is to factor the numerator and denominator:

$\displaystyle f(x)=-\frac{10x+7}{(2x-3)(x-7)}$

Now, we will assume the decomposition will take the form:

$\displaystyle -\frac{10x+7}{(2x-3)(x-7)}=\frac{A}{2x-3}+\frac{B}{x-7}$

Using the Heaviside cover-up method, we may find the value of $A$ by covering up the factor $(2x-3)$ on the left side, and evaluate what's left where $x$ takes on the value of the root of the covered up factor, i.e., $\displaystyle x=\frac{3}{2}$. Hence:

$\displaystyle A=-\frac{10\cdot\frac{3}{2}+7}{\frac{3}{2}-7}=4$

Likewise, we find the value of $B$ by covering up the factor $(x-7)$ on the left side, and evaluate what's left for $x=7$:

$\displaystyle B=-\frac{10\cdot7+7}{2\cdot7-3}=-7$

And so we conclude that:

$\displaystyle f(x)=\frac{-10x-7}{2x^2-17x+21}=\frac{4}{2x-3}-\frac{7}{x-7}$

To DW123 or any other guests viewing this topic, partial fraction decomposition is sometimes taught in pre-calculus, but usually in calculus as a means of integrating functions, so if you have other related questions, please feel free to post them in:

http://www.mathhelpboards.com/f21/

http://www.mathhelpboards.com/f10/
 
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FAQ: DW123's question at Yahoo Answers regarding partial fraction decomposition

What is partial fraction decomposition?

Partial fraction decomposition is a mathematical method used to simplify rational expressions by breaking them down into simpler fractions. This is achieved by expressing the rational expression as a sum of its individual components, where each component has a denominator of a linear or quadratic polynomial.

Why is partial fraction decomposition useful?

Partial fraction decomposition is useful in several mathematical calculations, particularly in integration and solving differential equations. It allows us to simplify complex rational expressions into smaller, more manageable components, making it easier to perform calculations and solve problems.

How do I perform partial fraction decomposition?

To perform partial fraction decomposition, you first need to factor the denominator of the rational expression. Then, you set up an equation where the sum of the individual components equals the original rational expression. You can then solve for the unknown coefficients using algebraic methods.

Can all rational expressions be decomposed using this method?

No, not all rational expressions can be decomposed using partial fraction decomposition. The expression must have a proper rational function, meaning the degree of the numerator is less than the degree of the denominator. If the degree of the numerator is greater than or equal to the degree of the denominator, other methods must be used.

Are there any limitations to partial fraction decomposition?

Partial fraction decomposition can only be used for rational expressions where the denominator can be factored into linear and/or quadratic polynomials. If the denominator contains higher degree polynomials or irreducible quadratic factors, other methods must be used to simplify the expression.

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