Question

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

Short answer

Question: Resolve the expression \(\frac{ax-b}{(x+1)^{2}}\) into partial fractions: Answer: \(\frac{ax-b}{(x+1)^{2}} = \frac{a}{x+1}\)

Step-by-step solution

Identify the correct form of partial fractions

We are given the expression \(\frac{ax-b}{(x+1)^{2}}\) and 4 different forms of partial fractions. In order to find the correct form, we need to rewrite the given expression in such a way that the denominators match the given forms. The given expression can be split into two fractions sharing the same denominator, which is \((x+1)^2\): $$\frac{ax-b}{(x+1)^{2}} = \frac{A}{x+1} + \frac{B}{(x+1)^{2}}$$ We'll call these constants A and B.

Write the expression using the correct form

Using the form from Step 1, we can write the expression as: $$\frac{ax-b}{(x+1)^{2}} = \frac{A}{x+1} + \frac{B}{(x+1)^{2}}$$ Now, we need to find the values of A and B.

Find the values of constants A and B

To find the values of A and B, we first clear the denominators by multiplying both sides by \((x+1)^2\): $$(x+1)^2 \left(\frac{ax-b}{(x+1)^{2}}\right) = (x+1)^2 \left(\frac{A}{x+1} + \frac{B}{(x+1)^{2}}\right)$$ This simplifies to: $$ax - b = A(x+1) + B$$ Now, we need to find the values of A and B such that the equation holds true for any value of x. We can do this by solving the system of equations using substitution or elimination method. We'll try substitution: - Choose x = -1 and substitute in the equation: $$a(-1) - b = A(-1+1) + B$$ $$-a - b = B$$ - Choose x = 0 and substitute in the equation: $$a(0) - b = A(0+1) + B$$ $$-b = A+B$$ Now solve this system of equations: Eliminate B: $$B = -a - b$$ $$-b = A - a - b$$ $$A = a$$ Substitute A back into one of the equations: $$-b = a - a - b$$ $$B = 0$$ Now we have the values of the constants A and B: $$A = a$$ $$B = 0$$ Finally, we can write the proper partial fraction for the original expression: $$\frac{ax-b}{(x+1)^{2}} = \frac{a}{x+1} + \frac{0}{(x+1)^{2}} = \frac{a}{x+1}$$ Thus, the correct answer is Option (1): $$\frac{ax-b}{(x+1)^{2}} = \frac{a}{x+1}$$

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The purpose of algebraic expressions is to represent real-world problems or situations in a mathematical form to analyze and solve them.

In the given example, \(\frac{ax-b}{(x+1)^{2}}\) is an algebraic expression that consists of variables a and x, a constant b, and is constructed using algebraic operations. To simplify such expressions or solve equations involving them, different algebraic techniques are used. One such technique is breaking down complex algebraic fractions, known as rational expressions, into simpler parts; this particular method is called partial fraction decomposition.

Rational Expressions

A rational expression is much like a fraction, but instead of integers, its numerator and denominator are algebraic expressions. Such fractions are common in higher-level maths and can look intimidating. However, they operate under the same principles as simple fractions. Simplifying rational expressions often involves factoring the numerators and denominators, finding common denominators, and reducing terms when possible.

In solving the exercise, we encountered a rational expression that needs to be decomposed into partial fractions. The rationale behind this process is to transform a complex rational expression into a sum or difference of simpler fractions, which are easier to integrate or differentiate if needed, or to solve. This method is vital in calculus, to simplify integration of rational functions.

Mathematical Problem-solving

The approach to mathematical problem-solving involves a series of steps that guide us from understanding the problem to arriving at a solution. In the context of the given exercise, this begins with identifying that the given complex fraction can be broken down into partial fractions.

After identification, the next step involves setting up an equation with unknown constants that represent the partial fractions which, when combined, should equal the original expression. The following action is to solve for these constants, which typically requires finding values for the variable that make the algebra easier (such as setting the variable to zero or choosing a value that simplifies one of the expressions to zero), and solving the resulting system of equations. These systematic steps allow students to tackle complex problems by breaking them down into manageable parts and provide an organized pathway toward the solution.