Question

Write the partial fraction decomposition for the expression. $$ \frac{7 x+5}{6\left(2 x^{2}+3 x+1\right)} $$

Short answer

The partial fraction decomposition for the given expression is \(\frac{7x + 5}{6(2x^2 + 3x + 1)}\). Because the quadratic term in the denominator cannot be further factorized, only one constant is obtained.

Step-by-step solution

Extract the factors of the quadratic term in the denominator

This step requires you to factorize the quadratic term. However, the quadratic term \(2x^2 + 3x + 1\) cannot be further factorized because it has only one unique root. So, we proceed without factorizing the denominator.

Express the fraction as partial fractions

A fraction can be expressed as the sum of its partial fractions. Since the denominator is a quadratic having only one root, we express the fraction in the following form: \(\frac{7x + 5}{6(2x^2 + 3x + 1)} = \frac{A}{2x^2 + 3x + 1}\)

Solve for the constant in the partial fraction

To solve for the constant \(A\), we multiply through by \(2x^2 + 3x + 1\), yielding \(7x + 5 = A\). As there is no x term in the equation, this is a valid standing equation that always holds true. Thus, we can solve for \(A\) by substituting the root of denominator (values of x that make denominator zero) and then solve for \(A\)