Question

Find the constants \(a, b, c\) and \(d\) respectively, if \(\frac{1}{x^{4}-x}=\frac{a}{x}+\frac{b}{x-1}+\frac{c x+d}{x^{2}+x+1}\). (1) \(-1,1 / 3,-1 / 2,-5 / 6\) (2) \(1,1 / 3,-1 / 2,5 / 6\) (3) \(1,-1 / 3,1 / 2,5 / 6\) (4) None of these

Short answer

Question: Provide the correct option for the following constants obtained from the equation \(\frac{1}{x^{4}-x}\): (1) -1, 1 / 2, -1 (2) 1, -1 / 3, 1 (3) -1, 1 / 3, -1 (4) 1, -1 / 2, 1 Answer: (3) -1, 1 / 3, -1

Step-by-step solution

Identify the terms

To solve for the constants, first identify the terms of the given equation that we will break down: \(\frac{1}{x^{4}-x}=\frac{a}{x}+\frac{b}{x-1}+\frac{cx+d}{x^{2}+x+1}\).

Clear the denominators

Multiply both sides of the equation by the common denominator \(x(x-1)(x^2+x+1)\) to eliminate the fractions: $$1 = a(x-1)(x^{2}+x+1) +bx(x^{2}+x+1) + (cx+d)x(x-1).$$

Expand and group

Expand the terms on the right side of the equation and group the like powers of \(x\) together: $$1 = (a + bx) (x^{3} - x^{2} + x - 1) + (cx+d) x(x-1).$$

Match the coefficients of like terms

To match the coefficients of the like terms, we need to compare both sides of the equation. From the given equation, we have the following: 1. Coefficient of \(x^3\): \(a + b - c = 0\) 2. Coefficient of \(x^2\): \(-a - b + d = 0\) 3. Coefficient of \(x\): \(a + b - d = 0\) 4. Constant term: \(-a = 1\)

Solve the system of equations

Now, we will solve the system of linear equations obtained in step 4. From the fourth equation, we have \(a = -1\). Now plug this value of \(a\) into the other equations and solve for \(b\), \(c\), and \(d\): 1. \(b - c = 1\) 2. \(d = 2\) 3. \(b + d = 1\) By solving this system, we obtain \(b = 1\), \(c = 3\), and \(d = -1\).

Conclusion

The constants \(a, b, c\), and \(d\) are \(a=-1, b=1, c=3,\) and \(d=-1\). Therefore the correct answer is option (3) \(-1,1 / 3,-1\).

Key concepts

Algebraic Fractions

Algebraic fractions are fractions where both the numerator and the denominator are algebraic expressions, i.e., expressions involving variables. They are an extension of arithmetic fractions but include polynomial expressions. For example, in the problem statement, \(\frac{1}{x^4-x}\) is an algebraic fraction since its denominator is a polynomial. Understanding algebraic fractions is crucial because they form the basis of many operations in algebra, such as simplification and solving equations.

• To work with algebraic fractions, finding a common denominator is often necessary to combine or compare them.
• We use the concept of partial fraction decomposition to express a complex fraction as a sum of simpler ones, each with potentially lower degree polynomials in the numerator and denominator.

When working with algebraic fractions, always be mindful of the equality sign and valid operations that can transform the equation while keeping it equivalent.

Polynomial Equations

Polynomial equations are equations in which all terms are polynomials. A polynomial is an expression consisting of variables, coefficients, and operations of addition, subtraction, multiplication, and non-negative integer exponents. In the given problem, the denominator \(x^4-x\) can be expanded and treated as a polynomial equation.

• The quintessential property of polynomial equations is that their solutions, or roots, closely relate to the factors of the polynomial.
• Polynomial equations can sometimes be factored into simpler polynomials, which provide insights into the roots or zeroes of the equation.

The strategy often involves expanding or breaking down polynomials into more manageable parts to solve equations or simplify expressions. This decomposition can uncover hidden relationships or solutions within the equation.

System of Linear Equations

A system of linear equations comprises two or more linear equations. The goal is to find the values of the variables that satisfy all equations simultaneously. In the exercise, matching coefficients leads to a system of linear equations that we solve to find the constants \(a, b, c,\) and \(d\).

• Each equation in the system represents a relationship between the variables, depicted as a line on a graph.
• The solution is where all lines intersect, representing values that satisfy all equations.

To solve a system of linear equations, methods such as substitution, elimination, or using matrices can be applied. For simple systems, like the one in our exercise, comparing and substituting directly may be the most efficient method.

Coefficient Comparison

Coefficient comparison is a technique used to identify unknown coefficients in algebraic equations. By equating the coefficients of corresponding powers of a variable on both sides of an equation, we establish relationships between the unknowns. In the problem, by comparing coefficients of each power of \(x\), we derived equations to solve for \(a, b, c,\) and \(d\).

• Start by ensuring each side of the equation is simplified and explicitly arranged by powers of the variable.
• The coefficients of like terms from each side of the equation are set equal to each other.

This technique simplifies the problem into a series of linear equations. These equations arise from the equality of the coefficients of like powers of \(x\) on both sides, allowing us to solve for unknowns in an orderly manner.