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Partial Fraction Decomposition of 1/(3x^2 + 4x + 1)
Published on October 27, 2024
Algebra
The rational function 1/(3x^2 + 4x + 1) is decomposed into partial fractions as (3/2)/(3x + 1) - (1/2)/(x + 1) by factoring the denominator and solving for the unknown constants using strategic values of x.
Question
partial fractions 1/(3x^2 + 4x + 1)
Answer
Let’s break down how to decompose the rational function
1. Factor the Denominator
First, we factor the quadratic in the denominator:
2. Set up the Partial Fractions
Since we have two distinct linear factors, we can express the rational function as the sum of two fractions with unknown constants:
3. Clear the Fractions
Multiply both sides of the equation by the original denominator to get rid of the fractions:
4. Solve for the Constants
We can solve for
-
Let x = -1: This eliminates
, allowing us to solve for : -
Let x = -1/3: This eliminates
, allowing us to solve for :
5. Write the Final Decomposition
Now that we know
Therefore, the partial fraction decomposition of