Question

Resolve \(\frac{4 x+3}{x^{3}-7 x-6}\) into partial fractions. (1) \(\frac{1}{4(x+1)}+\frac{3}{4(x-3)}-\frac{1}{(x+2)}\) (2) \(\frac{-1}{4(x+1)}+\frac{3}{4(x-3)}+\frac{1}{(x+2)}\) (3) \(\frac{1}{4(x+1)}-\frac{3}{4(x-3)}+\frac{1}{(x+2)}\) (4) \(\frac{1}{4(x+1)}-\frac{3}{4(x-3)}-\frac{1}{x+2}\)

Short answer

Question: Express the fraction \(\frac{4x + 3}{x^3 - 7x - 6}\) as a sum of partial fractions. Answer: \(\frac{4x + 3}{x^3 - 7x - 6} = \frac{3}{x - 3} + \frac{1}{4(x + 1)} - \frac{1}{x + 2}\)

Step-by-step solution

Factorize the denominator of the given fraction

Since the denominator is a cubic polynomial, we need to factorize it to apply the partial fractions method. By trying different factors, we find that: $$x^3 - 7x - 6 = (x - 3)(x + 2)(x + 1)$$

Express the fraction as a sum of partial fractions

Now, write the given function as the sum of simple fractions: $$\frac{4x + 3}{(x - 3)(x + 2)(x + 1)} = \frac{A}{x - 3} + \frac{B}{x + 1} + \frac{C}{x + 2}$$ Clear the denominators by multiplying both sides by \((x-3)(x+2)(x+1)\): $$4x + 3 = A(x + 1)(x + 2) + B(x - 3)(x + 2) + C(x - 3)(x + 1)$$

Solve for the constants A, B, and C

We need to find the constants A, B, and C. We can do that by substituting convenient values for x: 1. Let x = 3: \(4(3)+3=15=A(0)(5)\), so \(A=\frac{15}{5}=3\). 2. Let x = -1: \(4(-1)+3=-1=B(-4)(1)\), so \(B=\frac{-1}{-4}=\frac{1}{4}\). 3. Let x = -2: \(4(-2)+3=-5=C(-5)(-1)\), so \(C=\frac{-5}{5}=-1\). Now we have the values of \(A=3\), \(B=\frac{1}{4}\), and \(C=-1\).

Write the final expression for the partial fractions

Substitute the values A, B, and C back into the partial fractions expression: $$\frac{4x + 3}{x^3 - 7x - 6} = \frac{3}{x - 3} + \frac{\frac{1}{4}}{x + 1} + \frac{-1}{x + 2}$$ Simplify the expression to match the answer options: $$\frac{4x + 3}{x^3 - 7x - 6} = \frac{3}{x - 3} + \frac{1}{4(x + 1)} - \frac{1}{x + 2}$$ The correct answer is option (1): $$\frac{4x + 3}{x^3 - 7x - 6} = \frac{3}{x - 3} + \frac{1}{4(x + 1)} - \frac{1}{x + 2}$$

Key concepts

Polynomial Factorization

Factorization is a method to break down polynomials into simpler, irreducible factors. Understanding polynomial factorization is essential for solving algebraic problems, including the simplification of complex rational expressions through partial fractions.

For instance, in the exercise above, the denominator of the given fraction is a cubic polynomial, which we factorized into \(x - 3)(x + 2)(x + 1)\). To factorize polynomials, one can use various techniques such as synthetic division, the Rational Root Theorem, and grouping. The first step generally involves looking for obvious factors, and then using methods tailored to the specific polynomial's degree and form to find the remaining factors. The ability to factorize polynomials effectively is pivotal when decomposing algebraic fractions into more manageable pieces.

Mastering polynomial factorization allows students to simplify complex expressions and solve equations that are otherwise too cumbersome to work with. Through practice, students can learn to quickly identify common factoring patterns and apply the appropriate strategies.

Algebraic Fractions

Algebraic fractions are simply fractions where the numerator and/or the denominator contain algebraic expressions, usually including variables. In our exercise, for example, the algebraic fraction in question is \(\frac{4x+3}{x^3-7x-6}\). Algebraic fractions, much like their numerical counterparts, follow the same principles for operations such as addition, subtraction, multiplication, and division.

However, algebraic fractions often require additional steps to simplify or resolve, like finding a common denominator when adding or subtracting, or multiplying by the reciprocal in the case of division. These operations become more intuitive with the knowledge of factoring polynomials, as this helps to simplify or cancel out terms. Grasping the basic principles of algebraic fractions is indispensable for any student aiming to do well in algebra, particularly in solving equations and working with complex rational expressions.

Rational Expressions

Rational expressions are expressions that consist of a ratio of two polynomials. They are the algebraic equivalent of rational numbers (the ratio of two integers). The process of splitting such expressions into partial fractions, as done in our exercise, is a way to break down more complex rational expressions into simpler, more easily solved components.

The goal with partial fractions is to rewrite the complex rational expression into a sum or difference of simpler fractions, making calculus operations like integration or finding limits much more straightforward. The partial fraction decomposition method is vital when dealing with integration of rational expressions in calculus. Each type of denominator demands a corresponding form for its partial fraction; for linear factors, the form is \(\frac{A}{(ax+b)}\), and for irreducible quadratic factors, it's \(\frac{Ax+B}{(ax^2+bx+c)}\). By mastering rational expressions and partial fractions, students become well-equipped to handle a variety of problems in both algebra and calculus.