Problem 74 - Algebra for College Students

Question

Find the indicated value for each given rational expression, if possible. $$N(x)=\frac{x+3}{x^{3}-2 x^{2}-2 x-3}, N(3)$$

Step-by-Step Solution

Answer

The rational expression N(3) is undefined since division by zero is not allowed.

Step 1 - Substitute the value

Substitute the given value into the function. Here, you need to find the value of the function when x = 3. Substitute x = 3 into the expression for N(x): ewline $$N(3)=\frac{3+3}{3^{3}-2\times 3^{2}-2\times 3-3}$$

Step 2 - Simplify the numerator

Simplify the numerator of the rational expression. In the numerator, we have: ewline $$3 + 3 = 6$$

Step 3 - Simplify the denominator

Now, simplify the denominator. Calculate each term in the denominator individually: ewline $$3^{3} = 27$$, $$2 \times 3^{2} = 18$$, $$2 \times 3 = 6$$. ewline Then, substitute these values into the expression:ewline $$27 - 18 - 6 - 3$$

Step 4 - Complete the simplification

Now, perform the subtraction to simplify the denominator: ewline $$27 - 18 = 9$$ $$9 - 6 = 3$$ $$3 - 3 = 0$$. ewline Therefore, the denominator simplifies to 0.

Step 5 - Determine the value of the rational expression

Since the denominator of the rational expression is 0, the rational expression is undefined. This is because division by zero is not allowed in mathematics.

Key Concepts

Substitution Method

The substitution method is crucial for solving rational expressions where you need to find the value of the function at a specific point. You start by replacing the variable in the rational expression with the given value. For example, in our exercise, we were given the value of 3 for the variable x in the expression \( N(x) = \frac{x + 3}{{x^3 - 2x^2 - 2x - 3}} \). We then substitute 3 for x, resulting in: \( N(3) = \frac{3 + 3}{{3^3 - 2 \times 3^2 - 2 \times 3 - 3}} \). By substituting the value, we simplify and solve the problem step by step. The substitution sets the stage for more straightforward calculations.

Simplifying Expressions

Simplifying expressions involves breaking down the parts of the rational expression to their simplest forms. After substituting the value, we simplified the numerator and the denominator separately.
For our example, we first simplified the numerator: \(3 + 3 = 6\).
Then, we simplified the terms in the denominator: \(3^{3} = 27, 2 \times 3^{2} = 18, \text { and } 2 \times 3 = 6\)
Then, we performed the calculations in sequence to break down the denominator: 27 - 18 = 9, 9 - 6 = 3, and finally, 3 - 3 = 0.
Simplifying expressions effectively makes the operations more manageable and comprehensible.

Undefined Expressions

A rational expression becomes undefined when its denominator equals zero, and this concept is crucial in understanding the limitations of such expressions.
In our exercise, the denominator ended up as zero: \( N(3) = \frac{6}{{0}} \).
Because any number divided by zero is undefined, we cannot calculate \(N(3) \). This undefined nature holds significant importance in algebra and calculus, especially when working with rational expressions.
Keeping an eye out for when expressions become undefined helps avoid mathematical errors and ensures accurate computations.

Division by Zero

Division by zero is a fundamental math concept that states dividing any number by zero is undefined. In simple terms, you cannot divide something into zero parts. For our rational expression, the denominator was reduced to zero, hence: \( N(3) = \frac{6}{{0}} \).
Mathematics strictly prohibits such operations because they do not yield meaningful or finite results.
As seen in our exercise, encountering a zero denominator immediately tells us that the rational expression cannot be computed at that point. This is a crucial rule to remember – always check your denominator to avoid the mistake of division by zero.
Recognizing and understanding this concept will significantly aid in solving and simplifying rational expressions correctly.