Problem 41 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{5}{x^{2}-9 x+8}-\frac{3}{x^{2}-6 x-16} $$

Step-by-Step Solution

Answer

\(\frac{2x + 13}{(x-1)(x-8)(x+2)}\)

Step 1: Factor Denominators

Identify and factor the denominators of each fraction. The first denominator is \(x^2 - 9x + 8\), which factors into \((x-1)(x-8)\). The second denominator is \(x^2 - 6x - 16\), which factors into \((x-8)(x+2)\).

Step 2: Find the Least Common Denominator (LCD)

The least common denominator for the fractions is the product of the distinct factors: \((x-1)(x-8)(x+2)\).

Step 3: Rewrite Each Fraction with the LCD

Transform each fraction to have the common denominator \((x-1)(x-8)(x+2)\):- The first fraction: \(\frac{5}{(x-1)(x-8)} \rightarrow \frac{5(x+2)}{(x-1)(x-8)(x+2)}\)- The second fraction: \(\frac{3}{(x-8)(x+2)} \rightarrow \frac{3(x-1)}{(x-1)(x-8)(x+2)}\).

Step 4: Subtract the Fractions

Perform the subtraction on the numerators:\[\frac{5(x+2)}{(x-1)(x-8)(x+2)} - \frac{3(x-1)}{(x-1)(x-8)(x+2)} = \frac{5(x+2) - 3(x-1)}{(x-1)(x-8)(x+2)}\].

Step 5: Simplify the Numerator

Expand and simplify the expression in the numerator:- Expand: \(5(x+2) = 5x + 10\) and \(-3(x-1) = -3x + 3\).- Simplify: \(5x + 10 - 3x + 3 = 2x + 13\).The expression simplifies to \(\frac{2x + 13}{(x-1)(x-8)(x+2)}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that helps in breaking down complex expressions into simpler multiplicative components. It's like identifying the building blocks of a polynomial. Consider the polynomial from the exercise: \(x^2 - 9x + 8\). Our goal is to express this as a product of binomials. Here's a simple guideline to factor a quadratic polynomial:

• Identify the coefficients: In this case, 1, -9, and 8.
• Find two numbers that multiply to the constant term (8) and add to the linear coefficient (-9).
• The numbers -1 and -8 fit this criteria: \(-1 \times -8 = 8\) and \(-1 + (-8) = -9\).

Thus, \(x^2 - 9x + 8\) factors to \((x-1)(x-8)\). Similarly, for \(x^2 - 6x - 16\), find factors -8 and 2, resulting in \((x-8)(x+2)\). Factoring polynomials makes it much easier to find the least common denominator (LCD) and simplify expressions.

Least Common Denominator

The least common denominator (LCD) is crucial when adding or subtracting fractions with different denominators. It is the smallest expression divisible by each denominator's factors. Once you factor your denominators, finding the LCD becomes straightforward. The denominators in our exercise: \((x-1)(x-8)\) and \((x-8)(x+2)\), share \((x-8)\) as a common factor.

• Write down each unique factor: \((x-1)(x-8)(x+2)\) encapsulates both denominators.
• The LCD is the product of these unique factors.

Thus, the LCD of \((x^2 - 9x + 8)\) and \((x^2 - 6x - 16)\) is \((x-1)(x-8)(x+2)\). Using the LCD simplifies the operation of adding or subtracting the fractions because each is then expressed with the same denominator.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing the expression to its simplest form. After finding the LCD and rewriting each fraction with it, perform arithmetic operations on the numerators while maintaining the common denominator. In the exercise, we had:

• First fraction's numerator: \(5(x+2)\).
• Second fraction's numerator: \(3(x-1)\).

Subtract these numerators: \(5(x+2) - 3(x-1)\). Expand:

• \(5(x+2) = 5x + 10\)
• \(-3(x-1) = -3x + 3\)

Combine like terms:

• \(5x + 10 - 3x + 3 = 2x + 13\).

The simplified form is \(\frac{2x + 13}{(x-1)(x-8)(x+2)}\). Proper simplification ensures expressions are fully reduced, making them easier to interpret and use. Always look for opportunities to factor further or cancel matching terms whenever possible.