Problem 44 - Algebra and Trigonometry

Question

Determine \(A, B, C,\) and \(D\) in terms of \(a\) and \(b :\) $$\frac{a x^{3}+b x^{2}}{\left(x^{2}+1\right)^{2}}=\frac{A x+B}{x^{2}+1}+\frac{C x+D}{\left(x^{2}+1\right)^{2}}$$

Step-by-Step Solution

Answer

\(A = a, B = b, C = -a, D = -b\).

Step 1: Set Up the Equation

We need to find constants \(A, B, C,\) and \(D\) so that the equation \( \frac{a x^3 + b x^2}{(x^2+1)^2} = \frac{Ax + B}{x^2+1} + \frac{Cx + D}{(x^2+1)^2} \) holds true for all \(x\). Begin by combining the right side of the equation over a common denominator, which is \((x^2+1)^2\).

Step 2: Combine Right Side over Common Denominator

The equation becomes \( \frac{(Ax + B)(x^2 + 1) + (Cx + D)}{(x^2+1)^2} \). Expand the numerator: \( (Ax + B)(x^2 + 1) = Ax^3 + Ax + Bx^2 + B \). Add \((Cx + D)\) to get: \(Ax^3 + Bx^2 + Ax + B + Cx + D\).

Step 3: Equate Numerators

The expression from Step 2 must equal the left side's numerator: \(a x^3 + b x^2\). Therefore, equate the expanded numerator: \(Ax^3 + Bx^2 + (A+C)x + (B+D) = a x^3 + b x^2\).

Step 4: Match Coefficients

To solve for \(A, B, C,\) and \(D\), equate the coefficients on both sides of the equation:- From \(x^3\), we get \(A = a\).- From \(x^2\), we get \(B = b\).- From \(x\), we have \(A + C = 0\) implying \(C = -a\).- The constant term gives \(B + D = 0\) so \(D = -b\).

Step 5: Verify the Solution

Substitute \(A = a\), \(B = b\), \(C = -a\), and \(D = -b\) in the equation to check if it satisfies the equation for all \(x\). Upon verification, both the left and right sides have the same numerator, confirming the solution is correct.

Key Concepts

Rational Expressions

Rational expressions are pivotal in algebra and calculus, acting as a gateway to understanding complex relationships within mathematical equations. A rational expression is essentially a ratio of two polynomials.
For instance, in the given exercise, the left-hand side of the equation is the rational expression:

• \( \frac{a x^{3} + b x^{2}}{(x^{2} + 1)^{2}} \)

The numerator consists of the polynomial \( ax^{3} + bx^{2} \), while the denominator is \( (x^{2} + 1)^{2} \).
Understanding rational expressions involves simplifying or breaking them down into simpler fractions or sums of fractions.Partial fraction decomposition comes into play when you want to break a complex rational expression into a sum of simpler expressions, which can be easier to integrate or differentiate.
This method is particularly useful when the denominator is factored, as seen in the original exercise where the common denominator

• \( (x^2 + 1)^2 \)

allows us to decompose the fraction into simpler components based on the powers of the factors in the denominator.

Polynomial Equations

Polynomial equations form the backbone of rational expressions and their manipulations. In a polynomial equation like the one from the exercise, we deal with expressions where variables are raised to whole number powers and combined using addition, subtraction, and multiplication. The equation in our problem,

• \( a x^{3} + b x^{2} = Ax^{3} + Bx^{2} + (A+C)x + (B+D) \)

demonstrates how every polynomial is a sum of monomial terms.
Each term has a coefficient and a power of the variable, \( x \).
Understanding how to manipulate such equations requires familiarity with key operations such as expansion, factoring, and equating coefficients. Equating coefficients is critical for solving our problem.
This involves aligning terms from both sides of the equation based on their degrees, leading to individual equations that can be solved for unknowns such as \( A, B, C, \) and \( D \).
Beyond this, polynomials also allow us to systematically break down and comprehend the effects of each term separately in any given equation.

Numerator and Denominator Manipulation

The ability to manipulate numerators and denominators is crucial in simplifying rational expressions and solving for unknown constants. Let's take a closer look at how this works in the context of the original exercise.We started by bringing the expression on the right side of the equation to a common denominator. This involved multiplying the numerator \( (Ax + B) \) by \( (x^2 + 1) \), which is a key step in aligning all terms through common factors. This manipulation allowed us to form:

• \( (Ax + B)(x^2 + 1) = Ax^3 + Bx^2 + Ax + B \)

This is then added to the simpler term \( Cx + D \), resulting in a combined numerator:

• \( Ax^3 + Bx^2 + (A+C)x + (B+D) \)

To solve the problem, we needed to ensure this complex numerator mirrored the polynomial \( ax^3 + bx^2 \) from the original rational expression's left side. This step of equating and comparing helps in the actual determination of constants \( A, B, C, \) and \( D \) necessary for partial fraction decomposition. This strategy of multiplying terms and aligning like terms for comparison is a backbone technique of algebraic manipulation. Understanding it makes handling complex rational expressions significantly easier.