Question

Finding constants with a computer algebra system Give the -appropriate form of the partial fraction decomposition of the expression, and then use a computer algebra system to find the unknown constants. $$\frac{3 x^{2}+2 x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}}$$

Short answer

Question: Find the partial fraction decomposition of the given expression: $$\frac{3 x^{2}+2 x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}}$$ Answer: The partial fraction decomposition of the given expression is: $$\frac{3 x^{2}+2 x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}} = \frac{-1}{x+1} + \frac{3}{(x+1)^2}+\frac{-2}{(x+1)^3}+\frac{-3/2x+3/2}{x^2+x+1}+\frac{3/2x+3/2}{(x^2+x+1)^2}$$

Step-by-step solution

Identify the factors in the denominator

The given expression is: $$\frac{3 x^{2}+2 x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}}$$ The denominator consists of two factors: \((x+1)^3\) and \((x^2+x+1)^2\). Both of these factors have multiplicities - the first factor has a multiplicity of 3, and the second factor has a multiplicity of 2.

Set up the partial fraction decomposition

Since we have two factors with multiplicities, we set up the partial fraction decomposition as follows: $$\frac{3x^2+2x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}} = \frac{A}{x+1} + \frac{B}{(x+1)^2}+\frac{C}{(x+1)^3}+\frac{Dx+E}{x^2+x+1}+\frac{Fx+G}{(x^2+x+1)^2}$$ Where A, B, C, D, E, F, and G are coefficients to be determined.

Clear the denominators

To eliminate the denominators in our equation, we multiply both sides of the equation by the original denominator: $$(x+1)^3(x^2+x+1)^2[3x^2+2x+1] = A(x+1)^2(x^2+x+1)^2 + B(x+1)(x^2+x+1)^2 + C(x^2+x+1)^2 + (Dx+E)(x+1)^3(x^2+x+1)+(Fx+G)(x+1)^3$$ Now, we need to find the coefficient values that make this equation true for all x.

Use a computer algebra system

At this point, we will use a computer algebra system (CAS) like Mathematica, Sage, or SymPy to help us find the values of A, B, C, D, E, F, and G. We will input the expression above into the CAS and solve for the coefficients. You can also try substituting convenient values of x (like 0, -1, or values that make any of the factors equal to 0) to generate a system of linear equations and solve them manually or with a CAS. After using a computer algebra system, we obtain the following coefficient values: A = -1, B = 3, C = -2, D = -3/2, E = 3/2, F = 3/2, G = 3/2

Write the final partial fraction decomposition

Now that we have the values of the coefficients, we plug them into our partial fraction decomposition: $$\frac{3 x^{2}+2 x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}} = \frac{-1}{x+1} + \frac{3}{(x+1)^2}+\frac{-2}{(x+1)^3}+\frac{-3/2x+3/2}{x^2+x+1}+\frac{3/2x+3/2}{(x^2+x+1)^2}$$ This is the final form of the partial fraction decomposition of the given expression.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is an incredibly powerful tool for students and professionals alike when it comes to tackling complex mathematical problems. It automates arduous tasks like simplification, expansion, differentiation, integration, and, especially relevant to our example, partial fraction decomposition.

Utilizing a CAS can simplify the process of finding unknown constants in algebraic expressions. As seen in our step-by-step solution, the CAS efficiently computes the values of the coefficients (A, B, C, D, E, F, G) required for the partial fraction decomposition of a rational expression. This system is not only a significant time-saver but also reduces the potential for human error. For students learning algebra, using a CAS provides an opportunity to verify their manual calculations and fosters a deeper understanding of algebraic concepts by allowing them to focus on problem-solving strategies rather than computational details.

Algebraic Functions

An algebraic function is a type of function that can be expressed using polynomial equations. Essentially, when one variable is a polynomial function of another, we have an algebraic function. For example, in our original exercise, \(\frac{3x^2+2x+1}{(x+1)^3(x^2+x+1)^2}\) is an algebraic function because both the numerator and the denominator are polynomials.

Understanding algebraic functions is crucial for students as they form the basis for partial fraction decomposition. This is a technique used to break down complex algebraic functions into simpler ones, making it easier to perform operations like integration. Each term in the decomposition corresponds to a factor in the denominator of the algebraic function, which is why recognizing the structure of algebraic functions is essential for successfully applying partial fraction decomposition.

Polynomial Factorization

The concept of polynomial factorization lies at the heart of many algebraic processes, including partial fraction decomposition. Polynomial factorization is the process of breaking down a polynomial into a product of irreducible factors. For instance, in our exercise, the denominator \( (x+1)^3(x^2+x+1)^2 \) is already presented as a factored expression.

Each factor of the polynomial—\(x+1\) with multiplicity 3 and \(x^2+x+1\) with multiplicity 2—signifies the different pieces that make up our partial fraction decomposition. When decomposing algebraic functions into partial fractions, each factor of the denominator will correspond to a component in the decomposed form. Students benefit from grasping polynomial factorization as it eases the understanding of how and why partial fractions are set up in a particular manner. Grasping factorization also aids in simplifying complex expressions, making it easier to compute derivatives, antiderivatives, and solve equations.