Problem 21 - Algebra A Combined Function

Question

Find any numbers for which each rational expression is undefined. $$ \frac{x}{3 x^{2}+13 x+14} $$

Step-by-Step Solution

Answer

The expression is undefined for \( x = -\frac{7}{3} \) and \( x = -2 \).

Step 1: Understanding the Problem

A rational expression is undefined when its denominator is equal to zero. Therefore, our goal is to find the value(s) of \( x \) that make the denominator of \( \frac{x}{3x^2 + 13x + 14} \) equal to zero.

Step 2: Setting the Denominator Equal to Zero

To find where the expression is undefined, set the denominator equal to zero: \( 3x^2 + 13x + 14 = 0 \).

Step 3: Factoring the Quadratic

Factor the quadratic equation. Look for two numbers that multiply to \( 3 \times 14 = 42 \) and add up to 13. These numbers are 6 and 7. Thus, factor the quadratic as \( 3x^2 + 6x + 7x + 14 = 0 \).

Step 4: Applying the Factoring by Grouping

Group terms to factor by grouping: \( (3x^2 + 6x) + (7x + 14) = 0 \). Factor out the common terms: \( 3x(x + 2) + 7(x + 2) = 0 \).

Step 5: Extracting the Common Factor

Notice that \( (x + 2) \) is a common term in both groups, so factor it out: \( (3x + 7)(x + 2) = 0 \).

Step 6: Setting Each Factor to Zero

Set each factor equal to zero: \( 3x + 7 = 0 \) and \( x + 2 = 0 \). Solve these equations separately.

Step 7: Solving the First Equation

Solve \( 3x + 7 = 0 \): \( 3x = -7 \), then \( x = -\frac{7}{3} \).

Step 8: Solving the Second Equation

Solve \( x + 2 = 0 \): \( x = -2 \).

Step 9: Concluding the Solution

The values of \( x \) that make the denominator zero are \( x = -\frac{7}{3} \) and \( x = -2 \). These are the values where the expression is undefined.

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill in algebra that allows you to rewrite polynomial equations in a more manageable form. Quadratic equations often come in the form \( ax^2 + bx + c = 0 \). To factor these, you need to find two numbers that both multiply to the product of \( a \) and \( c \) and add up to \( b \). This process involves looking for patterns or using the trial and error method.
In the given problem, we dealt with the quadratic \( 3x^2 + 13x + 14 \). By finding the two numbers that multiply to 42 (the product of 3 and 14) and add up to 13, we identified the numbers 6 and 7. This is because \( 6 \times 7 = 42 \) and \( 6 + 7 = 13 \). Understanding this concept is key to proceeding with the method of factoring by grouping.

Denominator Zero

For a rational expression to be undefined, its denominator must be equal to zero. This is because dividing by zero is mathematically impossible, as it would not yield a finite or meaningful result. In rational expressions, such as \( \frac{x}{3x^2 + 13x + 14} \), finding where the denominator equals zero pinpoints where the expression is undefined.
In this specific case, by setting \( 3x^2 + 13x + 14 = 0 \), we determine the critical values of \( x \) that will lead the denominator to zero. Once these values are found, the expression becomes undefined at these points, ensuring no solutions exist when these conditions are met.

Solving Equations

Solving equations involves finding the values of variables that satisfy the equation. After factoring a quadratic, setting each factor equal to zero helps find these solutions. When you have an expression like \( (3x + 7)(x + 2) = 0 \), the solutions arise from each factor.
To solve \( 3x + 7 = 0 \), subtract 7 from both sides to get \( 3x = -7 \). Then, divide by 3 resulting in \( x = -\frac{7}{3} \). Similarly, for \( x + 2 = 0 \), subtract 2 from both sides to find \( x = -2 \). Each solution indicates a point where the original rational expression becomes undefined due to zero in the denominator.

Factoring by Grouping

Factoring by grouping is a useful method for breaking down more complex polynomials. This technique involves grouping terms in pairs and factoring out common factors. In the expression \( 3x^2 + 6x + 7x + 14 \), we group it as \( (3x^2 + 6x) + (7x + 14) \).
The next step is to factor out the greatest common factor (GCF) in each group. From the first group \( 3x(x + 2) \), and from the second group \( 7(x + 2) \). The common factor \( (x + 2) \) can then be factored out of the entire expression, resulting in \( (3x + 7)(x + 2) \). This factored form allows you to set each part equal to zero to solve the quadratic equation effectively.