Problem 19 - Algebra 1

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: Factor the denominator

The denominator can be factored using the factoring method. Looking for two numbers-multiplied equal to 15 (last term), and added equals to 8 (middle term). The numbers are 5 and 3. So, it can be written as \(x^{2}-8x+15 = (x-5)(x-3)\).

Step 2: Simplify the equation

Replace the denominator in the original expression for its factored form: \(\frac{5-x}{(x-5)(x-3)}\). Note that the numerator is equal to the negative of \(x-5\), so we write the negative sign outside \(-\). Hence, this becomes \(\frac{-(x-5)}{(x-5)(x-3)}\). Then, cancel out the common factor \((x-5)\) in the numerator and denominator.

Step 3: Write the simplified expression

After cancelling out the common factors, the simplified expression becomes \(\frac{-1}{x-3}\). This is the simplest form of the given expression.

Key Concepts

Factoring

Factoring in algebra involves breaking down a complex expression into simpler components that, when multiplied together, give back the original expression. This can be particularly useful for simplifying expressions and solving equations. Let's take a closer look using the exercise as an example. The exercise involves the expression \(x^2 - 8x + 15\).

To factor this expression, we need to find two numbers that multiply to get the constant term, 15, and add up to the coefficient of the middle term, which is 8. In this case, 5 and 3 are the numbers that fulfill these requirements since \(5 \cdot 3 = 15\) and \(5 + 3 = 8\).

Therefore, the expression \(x^2 - 8x + 15\) can be written as \((x - 5)(x - 3)\). Factoring is key because it helps identify possible simplifications, particularly when working with rational expressions.

Simplification

Simplification requires reducing an expression to its simplest form. This makes calculations easier and reduces complexity, aiding in clearer understanding. In the given exercise, once we have factored the denominator, we can proceed to simplify the expression.

The expression initially is \(\frac{5-x}{x^2-8x+15}\). After factoring the denominator, as demonstrated in the previous section, we replace it to get \(\frac{5-x}{(x-5)(x-3)}\). Notice that the numerator, \(5-x\), can be rewritten as \(-(x-5)\).

Thus, the expression becomes \(\frac{-(x-5)}{(x-5)(x-3)}\). By canceling out the common factor \((x-5)\) present in both the numerator and the denominator, the expression simplifies to \(\frac{-1}{x-3}\). The simplified form represents the expression in the easiest form possible, without changing its value.

Rational Expressions

Rational expressions are similar to fractions but with polynomials in the numerator and/or the denominator. Simplifying rational expressions helps in performing operations like addition, subtraction, multiplication, or division more easily.

In this exercise, the expression \(\frac{5-x}{x^2-8x+15}\) is a rational expression. These kinds of expressions can often be simplified using factoring and cancellation, as we've seen. It's important to remember:

• The denominator should not be zero, since division by zero is undefined.
• After simplification, any restrictions on the variable (such as values that would make the denominator zero) should be noted.

For \(x\) in \((x-5)(x-3)\), it implies \(x eq 5\) and \(x eq 3\), which are restrictions based on the factor form of the denominator. Recognizing this makes it possible to work with rational expressions more smoothly and avoid undefined expressions.