Solve each problem. This rational function has two holes and one vertical asymptote.
$\text{ƒ}\left(x\right)=\frac{(x^3+7x^2-25x-175)}{(x^3+3x^2-25x-75)}$
What are the x-values of the holes?

Verified step by step guidance

1. Start by factoring both the numerator and the denominator of the rational function \( f(x) = \frac{x^3 + 7x^2 - 25x - 175}{x^3 + 3x^2 - 25x - 75} \). This will help identify common factors that cause holes and factors that cause vertical asymptotes.
2. Use polynomial factoring techniques such as factoring by grouping or synthetic division to factor the numerator \( x^3 + 7x^2 - 25x - 175 \) into simpler polynomial factors.
3. Similarly, factor the denominator \( x^3 + 3x^2 - 25x - 75 \) into its polynomial factors using the same methods.
4. Identify the common factors between the numerator and denominator. Each common factor corresponds to a hole in the graph of the function. The x-values that make these common factors zero are the x-values of the holes.
5. Set each common factor equal to zero and solve for \( x \) to find the exact x-values where the holes occur.

Key Concepts

Rational Functions and Their Simplification

A rational function is a ratio of two polynomials. Simplifying the function by factoring numerator and denominator helps identify common factors, which correspond to holes in the graph where the function is undefined but can be simplified.

Holes in Rational Functions

Holes occur at x-values where both numerator and denominator have a common factor that cancels out. These points are not in the domain, but the function approaches a finite value, unlike vertical asymptotes where the function grows without bound.

Vertical Asymptotes

Vertical asymptotes occur at x-values where the denominator is zero but the numerator is not zero, causing the function to approach infinity or negative infinity. Identifying these helps distinguish them from holes in the graph.