Problem 40 - Algebra 1: Concepts and Skills

Question

Simplify the expression if possible. $$ \frac{5-x}{x^{2}-8 x+15} $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{x-3}\).

Step 1: Factorize the Denominator

Begin by factorizing the quadratic expression, \(x^2-8x+15\). The factors of the quadratic equation are \(x-5\) and \(x-3\). This makes the expression to be \(\frac{5-x}{(x-5)(x-3)}\).

Step 2: Simplify the integrand

Next, rewrite \(5 - x\) in the numerator as \(-(x - 5)\) to match with one of the factors in the denominator. This simplifies the expression to \(-\frac{x-5}{(x-5)(x-3)}\).

Step 3: Simplify the Expression

Finally, cancel out the common factor \((x-5)\) in both the numerator and denominator to simplify the expression to \(-\frac{1}{x-3}\).

Key Concepts

Factoring Quadratics

When working with quadratic expressions, especially when they appear in rational expressions, factoring becomes a valuable tool. A quadratic expression generally has the form \(ax^2 + bx + c\). The key to simplifying them is to factorize them into two binomials.To factor a quadratic expression like \(x^2 - 8x + 15\), we need to find two numbers that both multiply to give the constant term, \(c\), which is 15 in this case, and add up to give the middle coefficient, \(b\), which is -8.Consider the expression \(x^2 - 8x + 15\):

• What multiplies to 15? (Options include 1 & 15, 3 & 5, -3 & -5, etc.)
• Which of these pairs add up to -8? (The answer is -3 and -5)

Thus, the expression can be factored into \((x-5)(x-3)\). Factoring simplifies the work and is a crucial first step in dealing with rational expressions.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions is much like working with numerical fractions. The goal is to simplify it into its simplest form.In this exercise, we have the rational expression \(\frac{5-x}{x^2-8x+15}\). After factoring the denominator, it becomes \(\frac{5-x}{(x-5)(x-3)}\). To allow for further simplification, especially cancelation, aligning the form of the numerator with one of the factors in the denominator becomes crucial.This is done by rewriting \(5-x\) as \(-(x-5)\), which reformed the expression into \(-\frac{x-5}{(x-5)(x-3)}\). This new format reveals common factors that can be canceled, which is a fundamental step when working with rational expressions.

Canceling Common Factors

Canceling common factors is a fundamental simplification strategy when dealing with fractions, and it works much the same way with rational expressions.Once we have rewritten the expression \(-\frac{x-5}{(x-5)(x-3)}\), we can see that \(x-5\) appears both in the numerator and denominator. Canceling them involves removing these identical terms from both parts of the fraction.

• This process relies on the property that any non-zero quantity divided by itself is 1.
• It is imperative to only cancel terms when they are multiplied by others and not when they are separated by addition or subtraction.

After canceling \(x-5\), the simplified expression becomes \(-\frac{1}{x-3}\). This approach condenses the rational expression to its simplest and most revealing form, aiding in further computations or evaluations.