Question

Resolve \(\frac{3 x-5}{x^{2}+3 x+2}\) into partial fractions. (1) \(\frac{7}{x+2}-\frac{5}{x+1}\) (2) \(\frac{-8}{x+2}-\frac{11}{(x+1)}\) (3) \(\frac{11}{x+2}-\frac{8}{x+1}\) (4) \(\frac{7}{x+2}+\frac{5}{x+1}\)

Short answer

Answer: The correct partial fraction decomposition of the given fraction is \(\frac{-8}{x+1} + \frac{11}{x+2}\).

Step-by-step solution

Factor the Denominator

First, factor the denominator \(x^2 + 3x + 2\) into \((x + 1)(x + 2)\).

Write the Given Fraction as a Sum of Two Simpler Fractions

Let \(\frac{3x - 5}{(x + 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2}\).

Clear the Denominators

Multiply both sides of the equation by \((x + 1)(x + 2)\) to clear the denominators: \(3x - 5 = A(x + 2) + B(x + 1)\).

Match the Coefficients

For the equation in Step 3 to be true for all x, the coefficients on the left and right sides must be equal. So, we have two equations: (1) \(A + B = 3\) (coefficients of x) (2) \(2A + B = -5\) (constant terms)

Solve the System of Equations

We will now solve the system of equations obtained in Step 4: From (1), we get \(B = 3 - A\). Substituting this in (2), we have \(2A + (3 - A) = -5\). Therefore, \(A = -8\), and \(B = 3 - (-8) = 11\).

Write the Partial Fraction Decomposition

Substitute the values of \(A\) and \(B\) found in Step 5 into the general form of the partial fraction decomposition we wrote in Step 2: \(\frac{-8}{x + 1} + \frac{11}{x + 2}\). So, the correct form of the given fraction in partial fractions is: \(\boxed{\frac{-8}{x+1} + \frac{11}{x+2}}\), which corresponds to option (2).

Key concepts

Polynomial Factorization

Factoring polynomials is an essential step in resolving rational expressions into partial fractions. In this exercise, the denominator of the expression \(\frac{3x-5}{x^2+3x+2} \)is a quadratic polynomial of the form \(ax^2 + bx + c\). Here, it's necessary to express it as a product of simpler binomial factors. Factoring involves finding two numbers which multiply to give the constant term (here, \(+2\)) and add to give the linear coefficient (here, \(+3\)).Breaking down the polynomial \(x^2 + 3x + 2\), we look for two numbers that multiply to 2 and add to 3. These numbers are 1 and 2, thus we can factor the expression into\((x + 1)(x + 2).\)This factorization simplifies the task of finding partial fractions, making it easier to handle rational expressions.

System of Equations

Once you've rewritten the original fraction in terms of partial fractions, you will typically get expressions for coefficients that require solving a system of linear equations. With partial fractions, this means identifying constants (e.g., \(A\) and \(B\) in this exercise) that, when adjusted through multiplication, align the polynomial degree and its terms accordingly. In our example, rewriting the fraction gave us\(\frac{3x - 5}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x+2}\)Next, multiply both sides by the common denominator \((x+1)(x+2)\)This removes the denominators, resulting in:\(3x - 5 = A(x + 2) + B(x + 1)\)Creating a system of equations through comparison between the constant and linear terms on both sides of the equation is crucial for pinpointing the values of \(A\) and \(B\). Through this, you equate the coefficients of like terms, generating separate equations for each independent term.

Coefficient Matching

Coefficient matching is the technique used to equate the coefficients of corresponding powers of \(x\) on both sides of an equation, ensuring the equality holds true. When expanding the equation \(3x - 5 = A(x + 2) + B(x + 1)\),we focus on isolating terms by polynomial degree. This results in two separate components:

• The coefficient of \(x\): \(A + B = 3\)
• The constant term: \(2A + B = -5\)

Using this method is crucial as it provides a structured pathway to solve for unknowns in algebraic fractions. These equations help form a reliable system to resolve such identities and extract the exact coefficients under given conditions.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying and decomposing these expressions is essential in algebra to facilitate easier operations and solutions, particularly when integrating or solving differential equations. In this specific problem, the process of resolving a rational expression into partial fractions includes expressing \(\frac{3x - 5}{x^2 + 3x + 2}\)in simpler forms as a sum of fractions with the simpler denominators found after factorization.This decomposition not only makes further manipulation easier but also allows for an intuitive approach to calculus applications like integration. A well-structured partial fraction facilitates evaluating such expressions especially when dealing with calculus problems where direct computations can be cumbersome.Understanding this decomposition brings clarity and simplicity, transforming complex algebraic fractions into manageable parts.