Problem 25 - Calculus for Biology and Medicine

Question

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=-\frac{10}{3 x^{2}+8 x-3} $$

Step-by-Step Solution

Answer

The partial-fraction decomposition is \( -\frac{3}{3x-1} + \frac{1}{x+3} \).

Step 1: Factor the Denominator

Begin by factoring the quadratic expression in the denominator of the function. The given expression is \(3x^2 + 8x - 3\). We need to find two numbers that multiply to \(-3 \times 3 = -9\) and add to \(8\). These numbers are \(9\) and \(-1\). Thus, \(3x^2 + 8x - 3\) can be rewritten and factored into \((3x - 1)(x + 3)\). This gives us the function: \(f(x) = -\frac{10}{(3x - 1)(x + 3)}\).

Step 2: Setup Partial-Fraction Decomposition

The function can now be expressed as a sum of partial fractions. Since the denominator is comprised of two linear factors, we can set up the partial fraction as:\(-\frac{10}{(3x - 1)(x + 3)} = \frac{A}{3x - 1} + \frac{B}{x + 3}\).

Step 3: Clear the Denominator

Multiply both sides by \((3x - 1)(x + 3)\) to eliminate the denominator resulting in: \(-10 = A(x + 3) + B(3x - 1)\).

Step 4: Expand and Collect Like Terms

Expand the right-hand side, which gives: \(Ax + 3A + 3Bx - B\). Collect like terms to get: \((A + 3B)x + (3A - B)\). The equation to solve is now \(-10 = (A + 3B)x + (3A - B)\).

Step 5: Solve the System of Equations

Set the coefficients of like terms equal to each other to solve for \(A\) and \(B\). This gives the system: \(A + 3B = 0\) and \(3A - B = -10\). Solve these simultaneous equations to find \(A = -3\) and \(B = 1\).

Step 6: Write the Partial-Fraction Decomposition

Substitute \(A\) and \(B\) back into the partial fraction form to get \(-\frac{10}{(3x - 1)(x + 3)} = \frac{-3}{3x - 1} + \frac{1}{x + 3}\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions can seem tricky, but with practice, it becomes manageable. A quadratic expression is generally in the form of \(ax^2 + bx + c\). To factor it, you need to find two numbers that multiply to \(ac\) and add to \(b\). This step often involves trial and error along with some intuition.

Let's look at the quadratic expression from the problem: \(3x^2 + 8x - 3\).

• First, compute the product of \(a\) and \(c\), which is \(3 \times (-3) = -9\).
• Next, identify two numbers that multiply to \(-9\) and add to \(8\). These numbers are \(9\) and \(-1\).
• Rewrite the middle term using these two numbers: \(3x^2 + 9x - x - 3\).
• Group and factor by grouping: \[(3x^2 + 9x) + (-x - 3)\].
• Factor out common terms: \[3x(x + 3) - 1(x + 3)\].
• Now factor by using the common binomial factor: \[(3x - 1)(x + 3)\].

This breaking down into factors makes it much easier to work in other mathematical contexts like partial fraction decomposition.

Linear Factors

When dealing with quadratic expressions, factoring them into linear factors is often the goal. Linear factors refer to polynomials of degree one, usually expressed as \(ax + b\). They are simpler to handle mathematically and provide an easier way to approach problems like partial fraction decomposition.

In the expression \(3x^2 + 8x - 3\), we faced the task of extracting linear factors. Once factored, it turned into \((3x - 1)(x + 3)\), each a linear factor.

• 3x - 1: This factor implies a simple line with a slope of 3 and a y-intercept of -1.
• x + 3: This too represents a line, with a slope of 1 and a y-intercept of 3.

Factoring into linear factors is crucial as it simplifies the expression, turning complex quadratic problems into more manageable linear ones. It allows further techniques such as partial fraction decomposition to be applied more straightforwardly.

Systems of Equations

Systems of equations come into play when solving for unknowns in multiple equations simultaneously. They often appear when breaking complex expressions into simpler parts, as seen in partial fraction decomposition.

In solving our problem, once we expressed the fraction with unknowns \(A\) and \(B\): \(-\frac{10}{(3x - 1)(x + 3)} = \frac{A}{3x - 1} + \frac{B}{x + 3}\), we needed to equate the numerators after clearing the denominators.

• The resulting equation \(-10 = A(x + 3) + B(3x - 1)\) expanded to \((A + 3B)x + (3A - B)\).
• By equating the coefficients of terms separately, we set up a system: \(A + 3B = 0\) and \(3A - B = -10\).
• Solving these simultaneously, we found \(A = -3\) and \(B = 1\).

The solutions \(A = -3\) and \(B = 1\) allow us to write the final partial-fraction decomposition. Mastery of systems of equations is fundamental, as it’s widely used in algebra to connect different parts of an equation to find unknown variables.