Question

If \(\frac{x}{(x-1)\left(x^{2}+1\right)^{2}}=\frac{P}{x-1}+\frac{Q x+R}{x^{2}+1}+\frac{S x+T}{\left(x^{2}+1\right)^{2}}\), then \(P+Q-R-S+T=\) ____. (1) \(5 / 4\) (2) \(3 / 2\) (3) \(9 / 7\) (4) \(8 / 9\)

Short answer

Question: Determine the sum of the constants P, Q, R, S, and T, from the partial fraction decomposition of the given expression: \(\frac{x}{(x-1)\left(x^{2}+1\right)^{2}}\). Answer: (2) \(\frac{3}{2}\).

Step-by-step solution

Write down the given expression

We have the following expression: \[ \frac{x}{(x-1)\left(x^{2}+1\right)^{2}}=\frac{P}{x-1}+\frac{Qx+R}{x^{2}+1}+\frac{Sx+T}{\left(x^{2}+1\right)^{2}} \]

Multiply both sides by the common denominator

We will multiply both sides by the common denominator, \((x-1)(x^2+1)^2\), to simplify the equation: \[ x = P(x-1)(x^2+1)^2 + (Qx+R)(x^2+1)(x-1) + (Sx+T)(x-1) \]

Simplify and expand the expression

Simplify and expand the equation we have: \[ x = P\left(x^3(x-1) + 2x^2(x-1) + (x-1)\right) + (Qx^3 + Qx^2 - Qx + Rx^2 + R - Rx) + (Sx^2 - Sx + Tx - T) \]

Collect similar terms

Collecting the similar terms, we obtain: \[ x = x^4(P) + x^3(P+Q-R-S) + x^2(2P+P+Q+R+T) + x(-P-Q-R+S+T) - P \]

Compare coefficients

Since this equation is valid for all \(x\), the coefficients of the corresponding powers of \(x\) on both sides must be equal. From the equation, we have: \[ P = 0,\quad P + Q - R - S = 1,\quad 2P + P + Q + R + T = 0,\quad -P - Q - R + S + T = 0,\quad - P = 0 \]

Solve the system of equations

Because \(P = 0\), the system of equations becomes: \[ Q - R - S = 1,\quad Q + R + T = 0,\quad - Q - R + S + T = 0 \] Now, we add the first and the third equation to get rid of \(S\), obtaining: \[ 2Q - 2R + T = 1 \] We can now rewrite this equation as: \[ T = 2R - 2Q + 1 \] Plug this into the second equation to solve for \(Q\): \[ Q + R + (2R - 2Q + 1) = 0 \] \[ 2R - Q = -1 \] Now, we can solve for \(R\) from the first equation: \[ R = Q + S + 1 \] Plug this into the second equation to solve for \(Q\) and \(S\): \[ 2(Q + 1) - Q = -1 \] \[ Q = -3 \] Now we can find \(R\) and \(S\) using the equations above: \[ R = -3 + S + 1 \] \[ S = 1 \] Finally, we have \(T = 2R - 2Q + 1 = 2(-2) - 2(-3) + 1 = 5\).

Calculate the sum P + Q - R - S + T

We now want to determine the sum: \[ P + Q - R - S + T = 0 + (-3) - (-2) - 1 + 5 = \boxed{\frac{3}{2}} \] The correct answer is \((2) \frac{3}{2}\).

Key concepts

Rational Functions

Rational functions are an essential topic in mathematics, particularly in algebra and calculus. They are expressions that consist of two polynomials, one in the numerator and the other in the denominator. These functions are expressed in the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). The domain of rational functions excludes any values that make the denominator zero, as division by zero is undefined.
Understanding rational functions is crucial because they model various real-world behaviors and systems, such as electrical circuits or population dynamics. They help us understand behavior at infinity and at points of discontinuity, which are key when studying asymptotic behavior and limits.
Some important aspects to remember about rational functions include:

• Vertical Asymptotes: Occur where the denominator \( Q(x) \) is zero, showing where the function grows indefinitely.
• Horizontal Asymptotes: Indicate the behavior of the function as \( x \) approaches infinity.
• Zeros of the Function: These are the values of \( x \) that make the numerator zero, resulting in the function outputting zero.

Being comfortable with these characteristics aids in graphing and analyzing rational functions effectively.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, involve fractions where the numerator and/or the denominator are algebraic expressions rather than simple numbers. Understanding how to manipulate these fractions is a key step toward mastering algebra and solving more complex problems.
One common technique used with algebraic fractions is simplifying them, which is similar to simplifying numerical fractions. You attempt to factor both the numerator and the denominator, cancelling out any common factors. This simplifies the expression to its simplest form. Simplification is crucial because it makes further calculations, such as addition, subtraction, multiplication, or division, straightforward and less error-prone.
When working with algebraic fractions, the following steps are essential:

• Factor the Numerator and Denominator: Look for factors common to both and cancel them out.
• Determine the Least Common Denominator (LCD): If you're adding or subtracting fractions, find a common denominator to combine them.
• Check for Restrictions: Identify values that make the denominator zero to avoid illegal operations.

Algebraic fractions help break down complex polynomial expressions, especially in partial fraction decomposition, which involves expressing a complex rational function as a sum of simpler fractions.

Polynomial Equations

Polynomial equations are equations that involve a polynomial expression set equal to zero. These equations can vary in degree, from linear (degree 1) to quadratic (degree 2), cubic (degree 3), and beyond. Each degree involves different techniques for solving. For instance, linear equations use simple algebraic manipulations, while quadratic equations often require factoring, completing the square, or applying the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Higher-degree polynomial equations may need methods like polynomial division or synthetic division, and may involve finding roots through factoring or using numerical methods. Understanding polynomial equations is vital for studying algebra, as they form the backbone of many mathematical concepts and applications, such as in calculus when finding critical points of functions.
Key points to remember for polynomial equations include:

• Degree: The highest exponent in the polynomial, guiding the number of potential solutions.
• Roots: Solutions of the equation, where the polynomial equals zero.
• Factoring: Breaking down the polynomial into simpler components that can be solved more easily.

In partial fraction decomposition, understanding polynomials is crucial because it helps simplify a rational function into more manageable parts, which is useful in integration and other mathematical operations.