Problem 50 - Elementary and Intermediate Algebra

Question

Simplify. See Example 4. $$ \frac{x^{2}-10 x+21}{x^{2}-3 x} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{x - 7}{x} \) with restrictions \( x \neq 0 \) and \( x \neq 3 \).

Step 1: Factor the Numerator

To begin simplifying the expression \( \frac{x^{2}-10x+21}{x^{2}-3x} \), we first factor the numerator \( x^{2} - 10x + 21 \). This can be factored into \((x - 3)(x - 7)\) because these two numbers multiply to 21 and add to -10.

Step 2: Factor the Denominator

Next, we factor the denominator \( x^{2} - 3x \). We can factor out an \(x\), resulting in \( x(x - 3) \), because both terms in the expression contain \(x\).

Step 3: Simplify the Fraction

Now that we have factored both the numerator and the denominator, the expression becomes \( \frac{(x - 3)(x - 7)}{x(x - 3)} \). We can cancel the common factor of \( (x - 3) \) from the numerator and the denominator, resulting in \( \frac{x - 7}{x} \).

Step 4: State the Simplified Expression

The simplified form of the original expression is \( \frac{x - 7}{x} \). It is important to note the restriction that \( x eq 0 \) and \( x eq 3 \) to avoid division by zero.

Key Concepts

Factoring Polynomials

When working with rational expressions, factoring polynomials is often the first step. This process involves expressing a polynomial as a product of its simpler components. In our example, we had to factor both the numerator and the denominator. For the numerator, \( x^2 - 10x + 21 \), we looked for two numbers that multiply to 21 and add to -10. These numbers are -3 and -7, so the factored form is \((x - 3)(x - 7)\).
Moving on to the denominator, \( x^2 - 3x \), we notice both terms have a common factor of \(x\). Factoring out \(x\), we get \(x(x - 3)\). By breaking down each part into its factors, we can more easily simplify the entire expression.
Factoring makes it simpler to identify and eliminate common terms, simplifying the rational expression significantly.

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form, making calculations or further operations easier. After factoring both the numerator and the denominator, the expression \( \frac{(x - 3)(x - 7)}{x(x - 3)} \) is ready to be simplified. Now, we can observe that the factor \((x - 3)\) appears in both the numerator and the denominator.
By canceling out this common factor, the expression becomes \( \frac{x - 7}{x} \).
Remember:

• Cancel only when the terms are multiplied.
• Never cancel terms across an addition or subtraction sign.

This process of simplifying makes the fraction more manageable and highlights any restrictions on the variable.

Restriction of Variables

In rational expressions, it's critical to consider the restriction of variables to avoid undefined scenarios, particularly division by zero. While simplifying \( \frac{x - 7}{x} \), we need to identify where the original expression would be undefined.
From the original expression \( \frac{x^2-10x+21}{x^2-3x} \), we find restrictions by setting the denominator equal to zero: \( x(x - 3) = 0 \).
Solving this, we determine \( x = 0 \) and \( x = 3 \) would cause the denominator to be zero, making the expression undefined. Thus, \( x eq 0 \) and \( x eq 3 \) are restrictions.
Reporting these restrictions ensures clarity when simplifying as it defines valid values that \(x\) cannot take, preserving the mathematical integrity of the expression.