Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Write the partial fraction decomposition for the expression. $$ \frac{3 x^{2}-2 x-5}{x^{3}+x^{2}} $$

Step-by-Step Solution

Answer

\(\frac{3x-5}{x^{2}}\)

Step 1: Factor the Denominator

The denominator of the provided expression can be factored using the common factor rule. This rule states that if two terms share a common factor, this factor can be taken out from these terms. The denominator \(x^{3}+x^{2}\) can be factored to be \(x^{2}(x+1)\).

Step 2: Decompose into Partial Fractions

A partial fraction decomposition of the fraction is done by expressing the original fraction as a sum of simpler fractions. The numerator of the original fraction dictates the numerator of the simpler fractions. Given that the denominator of the included expression has been factored into \(x^{2}(x+1)\), the expression would be decomposed into \(\frac{Ax+B}{x^{2}} + \frac{C}{x+1}\) where A, B and C are constants which we will calculate in the next steps.

Step 3: Equate coefficients and Solve for A, B and C

The right hand side should be identical to the original function. Consequently, multiplying out the right hand side will give \((Ax+B)(x+1) + Cx^{2}\) which simplifies to \(Ax^{2} + Bx + Ax + B + Cx^{2}\), then \(Ax^{2} + Cx^{2} + (A+B)x + B\), and finally \((A+C)x^{2} + (A+B)x + B\). On the left hand side, the coefficients of the polynomial are \(3x^{2} - 2x - 5\). By equating the coefficients on both sides, we can find the values for A, B and C. These equations will be: 1. A + C = 3, 2. A + B = -2, 3. B = -5. From 3., we deduce that B = -5. By substituting B into equation 2., we find A = -2 - B = -2 - (-5) = 3. Finally, by substituting A into equation 1., we find C = 3 - A = 3 - 3 = 0

Step 4: Substitute A, B and C into the partial fractions

Substitute the values A=3, B=-5, and C=0 into the partial fractions \(\frac{Ax+B}{x^{2}} + \frac{C}{x+1}\) to give us \(\frac{3x-5}{x^{2}} + \frac{0}{x+1}\). Since \(\frac{0}{x+1}\) is essentially 0, this term will drop out, leaving our final answer as \(\frac{3x-5}{x^{2}}\).