Problem 29 - Algebra 1

Question

For what values of the variable is the rational expression undefined? $$\frac{x-3}{x^{2}+5 x-6}$$

Step-by-Step Solution

Answer

The values of x that make the rational expression undefined are x=-6 and x=1

Step 1: Identify the Denominator of the Rational Expression

The denominator of the given rational expression is \(x^{2}+5x-6\)

Step 2: Set the Denominator Equal to Zero

To find the values of x that make the expression undefined, set the denominator equal to zero: \(x^{2}+5x-6=0\)

Step 3: Solve the Equation

To solve the quadratic equation, we can use factoring. The quadratic \(x^{2}+5x-6\) can be factored into \((x+6)(x-1)=0\)

Step 4: Solve for X

Setting each factor equal to zero gives the solutions, x=-6 and x=1

Key Concepts

Undefined Expressions

In mathematics, a rational expression is considered undefined whenever its denominator equates to zero. This is because division by zero is not defined in the realm of real numbers. When you come across a rational expression, it's important to focus on the denominator. This will help you identify the values that make the expression undefined.

Here's how it works: suppose you have a rational expression, make sure to take the denominator part of it. Then, ask yourself for which values this denominator equals zero. Once you have these values, you know your expression will be undefined at these points.

In the exercise, the expression given was \(\frac{x-3}{x^2+5x-6}\). The denominator \(x^2+5x-6\) should not be zero for the expression to be defined.

Quadratic Equations

Quadratic equations appear in the form of \(ax^2 + bx + c = 0\). These types of equations are known for having degree 2, which means they will usually have two solutions or roots. Solving a quadratic equation can be done in several ways:

• Factoring the equation.
• Using the quadratic formula.
• Completing the square.

For the exercise's denominator, \(x^2 + 5x - 6\), factoring was used. Factoring often involves breaking down the quadratic expression into two binomials. This is possible when you can easily observe and separate the factors of the constant term, \(c\), that add up to the coefficient \(b\).

Upon successful factoring, the quadratic expression becomes \((x + 6)(x - 1) = 0\). This indicates that solving \(x^2 + 5x - 6 = 0\) leads us to identify the points where the expression is undefined.

Factoring Polynomials

Factoring is a process used in algebra that involves expressing a polynomial as a product of its factors. It can simplify expressions and solve equations effectively. Here's a basic approach to factoring polynomials:

• Look for a greatest common factor (GCF) among the terms.
• For a quadratic polynomial like \(x^2 + 5x - 6\), consider these steps:
  • Find two numbers whose product is the constant term, \(c\), and whose sum is the coefficient of the middle term, \(b\).
  • Rewrite the polynomial as a product of two binomials, based on these numbers.
• Verify by expanding the factored expression to ensure it equals the original polynomial.

For the exercise, you end up with \((x + 6)(x - 1)\) as the factored form of the quadratic polynomial \(x^2 + 5x - 6\). When each factor is set to zero, it reveals the solutions \(x = -6\) and \(x = 1\), which are crucial for finding undefined expressions.