Question

Decompose the following into partial fractions. \(\left(2 x^{2}+5 x-1\right) /\left(x^{3}+x^{2}-2 x\right)\)

Short answer

The partial fraction decomposition of the given rational function is: \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)} = \frac{1}{2x} + \frac{3x + 9}{2(x^2 + x - 2)}\)

Step-by-step solution

Factorize the Denominator

The denominator is given by \(x^3 + x^2 - 2x\). Let's factorize it. \(x^3 + x^2 - 2x = x(x^2 + x - 2)\) Now we have the denominator factorized, let's proceed to writing the partial fractions.

Write Partial Fractions

We have the given expression as: \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)}\) Now, let us represent this fraction as the sum of two partial fractions: \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)} = \frac{A}{x} + \frac{Bx + C}{x^2 + x - 2}\)

Find the Coefficients

By combining the partial fractions on the right-hand side, we have: \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)} = \frac{A(x^2 + x - 2) + (Bx + C)(x)}{x(x^2 + x - 2)}\) We can say that the numerators on both sides must be equal. Therefore: \(2x^2 + 5x - 1 = A(x^2 + x - 2) + (Bx + C)(x)\) Expanding the right-hand side, we get: \(2x^2 + 5x - 1 = (A + B)x^2 + (A + C)x - 2A\) Now let's equate the coefficients of the powers of x on both sides: 1. Coefficient of \(x^2\): \(2 = A + B\) 2. Coefficient of x: \(5 = A + C\) 3. Constant term: \(-1 = -2A\) Now, we will solve these three equations to find the values of A, B, and C.

Write the Final Expression

From the third equation, we find that \(A = \frac{1}{2}\). From the first equation, we find that \(B = 2 - A = \frac{3}{2}\). From the second equation, we find that \(C = 5 - A = \frac{9}{2}\). Therefore, the partial fraction decomposition of the given rational function is: \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)} = \frac{\frac{1}{2}}{x} + \frac{\frac{3}{2}x + \frac{9}{2}}{x^2 + x - 2}\) Alternative (simplified) representation: \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)} = \frac{1}{2x} + \frac{3x + 9}{2(x^2 + x - 2)}\)

Key concepts

Polynomial Factorization

To begin dealing with partial fraction decomposition, we first need to factorize the polynomial in the denominator. Polynomial factorization involves breaking down a complex expression into simpler, multiply-able parts.
The denominator in our case is given by the equation \(x^3 + x^2 - 2x\). A common technique for factoring polynomials like this is to first pull out the greatest common factor (GCF) from all terms.

• Here, the GCF is \(x\), because each term in the polynomial includes it as a factor.

Through extracting the GCF, we can express the polynomial as \(x(x^2 + x - 2)\). This transformation simplifies the problem significantly and sets the stage for partial fraction decomposition. Understanding factorization helps in simplifying expressions and solving equations more easily.

Coefficient Comparison

After factorization, the next step in decomposing into partial fractions involves coefficient comparison. This method helps us solve for unknowns that will allow us to rewrite a complex fraction into simpler parts.
Suppose you have broken your fraction down into terms: \(\frac{A}{x} + \frac{Bx + C}{x^2 + x - 2}\).

• Here, \(A\), \(B\), and \(C\) are coefficients we need to discover.
• Expand the right side by multiplying through the denominators.
• Set the expanded expression equal to the numerator of the original fraction \(2x^2 + 5x -1\).

Through this expansion, you compare and equate the coefficients of like terms that come with the same powers of \(x\). By solving the resulting system of equations, you adjust the coefficients \(A\), \(B\), and \(C\) to match those on the original side, ultimately simplifying your work with polynomials.

Rational Expressions

In mathematics, rational expressions are fractions where the numerator and the denominator are polynomials. These expressions are fundamentally important when solving problems involving partial fractions.

• They allow us to perform operations like addition, subtraction, multiplication, and division on polynomials.
• Partial fraction decomposition is often used to simplify complex rational expressions.

The given expression \(\frac{2x^2 + 5x -1}{x(x^2 + x - 2)}\) is a typical rational expression. Decomposing it into partial fractions does not change its value but represents it as a sum of simpler fractions.
Working with rational expressions often requires a solid understanding of algebraic rules and properties, such as factoring and simplifying expressions. Mastering these skills can greatly facilitate your math tasks.

Equation Solving

Equation solving is a core aspect of mathematics that involves finding unknown values that satisfy an equation. In the context of partial fractions, it involves finding coefficients that make two polynomials equal.
Consideration of partial fractions requires solving equations that arise from equating and simplifying the fractions obtained after the factorization stage.

• Solve the system of equations generated from coefficient comparison.
• This involves typical algebraic techniques such as substitution or elimination.

For example, in our case, the system to solve was:

• \(2 = A + B\)
• \(5 = A + C\)
• \(-1 = -2A\)

Solving these will give you \(A = \frac{1}{2}\), \(B = \frac{3}{2}\), and \(C = \frac{9}{2}\), allowing you to write the partial fractions. Developing these skills can help in efficiently tackling a wide range of mathematical problems involving equations.