Problem 54 - Intermediate Algebra

Question

Simplify each function. List any restrictions on the domain. $$ f(x)=\frac{5 x^{2}+50 x}{x^{5}+10 x^{4}} $$

Step-by-Step Solution

Answer

The simplified function is \(f(x) = \frac{5}{x^3}\), with domain restrictions \(x \neq 0\) and \(x \neq -10\).

Step 1 - Factor the Numerator

The numerator is given by the expression \(5x^2 + 50x\). To factor this expression, we can factor out the greatest common factor, which is \(5x\).\[5x^2 + 50x = 5x(x + 10)\]

Step 2 - Factor the Denominator

The denominator is given by \(x^5 + 10x^4\). To factor this expression, we can again factor out the greatest common factor, which is \(x^4\).\[x^5 + 10x^4 = x^4(x + 10)\]

Step 3 - Simplify the Function

Now, substitute the factored numerator and denominator into the function.\[f(x) = \frac{5x(x + 10)}{x^4(x + 10)}\]Cancel the common factor \((x+10)\) present in both the numerator and the denominator:\[f(x) = \frac{5x}{x^4}\]Simplify the function further:\[f(x) = \frac{5}{x^3}\]

Step 4 - Identify Domain Restrictions

The original function \(f(x) = \frac{5x^2 + 50x}{x^5 + 10x^4}\) has restrictions in its domain where the denominator equals zero.The denominator is \(x^4(x + 10)\), which equals zero when \(x^4 = 0\) or \(x + 10 = 0\), i.e., \(x = 0\) or \(x = -10\).Thus, the restrictions on the domain are \(x eq 0\) and \(x eq -10\).

Key Concepts

Simplifying Rational Functions

Rational functions are expressions that resemble fractions where both the numerator and the denominator are polynomials. Simplifying these functions means reducing them to their simplest form. By doing this, you make the function easier to work with. To simplify, identify and cancel out the common factors in the numerator and the denominator. For example, in the function \(f(x) = \frac{5x^2 + 50x}{x^5 + 10x^4}\), both the numerator and denominator have a common factor \((x + 10)\). We factor this out, allowing us to simplify to \(\frac{5}{x^3}\). When simplifying, always break down each part fully and double-check that all shared factors are correctly canceled. This ensures the function is completely simplified, helping avoid mistakes in calculations and making it more efficient for further use.

Domain Restrictions

Understanding domain restrictions in functions is crucial as it helps you know what values the input \(x\) can or cannot take. For rational functions, restrictions occur where the denominator equals zero, since division by zero is undefined. For instance, in the expression \(\frac{5x^2 + 50x}{x^5 + 10x^4}\), the denominator \(x^5 + 10x^4\) can be simplified to \(x^4(x + 10)\). This shows the denominator equals zero for \(x = 0\) and \(x = -10\). Hence, the domain of the simplified function \(\frac{5}{x^3}\) excludes these values. Knowing these restrictions prevents errors when inputting numbers and ensures you use the function correctly.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is particularly useful when simplifying rational functions. Taking the numerator \(5x^2 + 50x\) from our example, you can factor out the greatest common factor, which is \(5x\), leaving you with \(5x(x + 10)\). Similarly, the denominator \(x^5 + 10x^4\) can be factored as \(x^4(x + 10)\). By breaking down polynomials into these simpler multiplicative structures, you easily see the common factors, which can then be canceled out or used to simplify the function further. This step is key in both simplifying expressions and identifying any constraints related to domain restrictions.