Question

Resolve \(\frac{1}{x^{2}+x+42}\) into partial fractions. (1) \(\frac{1}{13(\mathrm{x}-6)}+\frac{1}{13(\mathrm{x}+7)}\) (2) \(\frac{1}{(x-7)}+\frac{1}{(x-6)}\) (3) \(\frac{1}{(x-7)}-\frac{1}{(x-6)}\) (4) \(\frac{1}{13(x-6)}-\frac{1}{13(x+7)}\)

Short answer

Based on the step-by-step solution, the question for a short answer could be: Question: Resolve the fraction \(\frac{1}{x^2 + x + 42}\) into partial fractions. Answer: \(\frac{1}{13(x-6)}-\frac{1}{13(x+7)}\)

Step-by-step solution

Factoring the Denominator

First, let's factor the denominator: \[x^2 + x + 42 = (x^2 + 7x - 6x + 42) = x(x + 7) - 6(x + 7) = (x - 6)(x + 7) \]

Set Up the Partial Fractions

Now that we have factored the denominator, we can set up the partial fractions: \[\frac{1}{(x - 6)(x + 7)} = \frac{A}{x - 6} + \frac{B}{x + 7}\]

Clear the Denominators

To find the values for A and B, we can clear the denominators by multiplying each side by (x - 6)(x + 7): \[1 = A(x + 7) + B(x - 6)\]

Solve for A and B

Now we can solve for A and B by choosing appropriate values of x that will eliminate either A or B: Let x = 6: \[1 = 13A \Longrightarrow A = \frac{1}{13}\] Let x = -7: \[1 = -13B \Longrightarrow B = -\frac{1}{13}\]

Rewrite the Partial Fractions

Now that we have the values for A and B, we can plug them into the partial fractions expression: \[\frac{1}{x^2 + x + 42} = \frac{1}{13(x - 6)} - \frac{1}{13(x + 7)}\] The correct answer is (4): \(\frac{1}{13(x-6)}-\frac{1}{13(x+7)}\)

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions, which are easier to work with, integrate, or differentiate. This is particularly useful when dealing with algebraic fractions that have polynomials in their denominators.

Here's how it works in a nutshell:

• First, factor the denominator of the rational expression as completely as possible.
• Then, write out a sum of fractions, called partial fractions, where each fraction has one of those factored denominators as its own denominator and an unknown constant in the numerator.
• Finally, determine the unknown constants by clearing the denominators and solving the resulting equations.

In our exercise, after factoring the denominator of \(\frac{1}{x^2 + x + 42}\), we found \(\frac{1}{(x - 6)(x + 7)}\). We then set up the partial fractions \(\frac{A}{x - 6} + \frac{B}{x + 7}\) and solved for the constants A and B to obtain the simplified expression.

Algebraic Fraction Simplification

The simplification of algebraic fractions makes them more manageable and comprehensible. Simplifying such fractions often involves reducing them to their lowest terms or, as we've seen with partial fraction decomposition, expressing a complex fraction as a sum of simpler ones. This can involve:

• Factoring polynomials in the numerator and denominator where possible.
• Cancelling common factors between the numerator and the denominator.
• Splitting a single complex fraction into multiple simpler fractions, as done with partial fraction decomposition.

For our given problem, we simplify the algebraic fraction by resolving it into partial fractions after factoring the denominator. This approach transforms a complex fraction with a quadratic denominator into a sum of fractions with linear denominators, which is a cleaner and more straightforward form for further mathematical processing.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of smaller polynomials (factors) that, when multiplied together, give the original polynomial. Factorization makes it easier to solve polynomial equations and to simplify algebraic expressions.

The steps typically include:

• Searching for a greatest common factor (GCF) among the terms.
• Looking for patterns such as the difference of squares, perfect square trinomials, or sum/difference of cubes.
• Applying methods such as grouping, the quadratic formula, or synthetic division if higher degree polynomials are involved.

In our exercise, the given polynomial \(x^2 + x + 42\) was factored into \(x - 6)(x + 7)\), using polynomial factorization techniques such as looking for a pair of numbers that multiply to the product of the constant term and the leading coefficient (42 * 1 = 42) and add to the middle coefficient (1). Correctly factoring the polynomial is crucial for the subsequent steps in partial fraction decomposition.