Problem 63 - Precalculus with Limits

Question

When writing the partial fraction decomposition of the expression \(\frac{x^{3}+x-2}{x^{2}-5 x-14},\) the first step is to divide the numerator by the denominator.

Step-by-Step Solution

Answer

The partial fraction decomposition will involve performing polynomial long division, rewriting the expression, factoring the denominator, and writing down the partial fractions using the factors obtained.

Step 1: Polynomial division

The first step is to divide \(x^{3}+x-2\) by \(x^{2}-5 x-14\). This step is necessary because the degree of the numerator is larger than the degree of the denominator. This is known as Polynomial Long Division.

Step 2: Writing the divided expression

Write down the expression resulting from the division in Step 1. This expression will be a polynomial plus a fraction where the degree of the numerator is less than the degree of the denominator.

Step 3: Factoring the denominator

Factorize \(x^{2}-5x-14\). The factors will be used in the partial fractions decomposition.

Step 4: Writing the partial fraction decomposition

Use the factors from Step 3 to write the partial fractions decomposition. Depending on the number of factors of the denominator, the expression will consist of a sum of terms which have the factors in the denominator and constants in the numerator.

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers. You work with expressions composed of variables raised to whole number powers and coefficients. Let's break this down:

• First, compare the leading term of the numerator with the leading term of the denominator.
• Divide those leading terms to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract it from the original numerator.
• Repeat this process with the new polynomial expression that results from the subtraction.

Continue until the degree of the remaining polynomial (remainder) is less than the degree of the divisor. At this point, you can express the original fraction as the sum of the polynomial quotient plus a proper fraction. This is crucial in writing partial fractions, as the quotient and remainder become components of the decomposed expression.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its linear and irreducible quadratic factors. It is a key step in partial fraction decomposition, especially for the denominator. Here's how you can factor a quadratic polynomial like \(x^2 - 5x - 14\):

• Look for two numbers that multiply to the constant term and add to the coefficient of the linear term. For \(x^2 - 5x - 14\), these numbers are \(-7\) and \(2\).
• Rewrite the polynomial as \((x-7)(x+2)\).

Once factored, these expressions create simpler denominators for decomposing the original rational expression into partial fractions. If the polynomial is higher degree, continue factoring until you simplify all terms.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable present in the polynomial expression. This concept helps you determine the approach to simplify or decompose expressions. Here's a closer look:

• The degree tells you how many solutions or roots the polynomial has, although some may be complex or repeated.
• In operations like polynomial division, comparing the degrees of the numerator and denominator guides you in appropriately reducing the expression.
• For proper rational expressions, the degree of the numerator must be lower than the degree of the denominator.

For example, in the polynomial \(x^3 + x - 2\), the degree is 3. Knowing the degree helps identify when partial fraction decomposition is viable, or if other algebraic techniques are needed.