(x^3-9)/(x^2+1)

4 min read Jun 17, 2024

Analyzing the Expression (x^3 - 9) / (x^2 + 1)

This expression represents a rational function, which is a function that can be expressed as the ratio of two polynomials. In this case, the numerator is x^3 - 9 and the denominator is x^2 + 1.

Simplifying the Expression

While we can't factor the denominator further, we can factor the numerator using the difference of cubes formula:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Applying this to our expression, we get:

(x^3 - 9) / (x^2 + 1) = (x - 3)(x^2 + 3x + 9) / (x^2 + 1)

This simplification might be helpful for certain operations, like finding the zeros of the function.

Finding the Zeros

To find the zeros of the function, we set the entire expression equal to zero and solve for x. This means the numerator must be equal to zero:

(x - 3)(x^2 + 3x + 9) = 0

We can see that x = 3 is one solution. The quadratic factor (x^2 + 3x + 9) has no real roots, as its discriminant (b^2 - 4ac) is negative. This means the function only has one real zero at x = 3.

Asymptotes

Rational functions can have vertical and horizontal asymptotes.

Vertical Asymptotes: These occur where the denominator is zero. In this case, the denominator (x^2 + 1) is never zero for any real value of x. Therefore, there are no vertical asymptotes.
Horizontal Asymptotes: We can find the horizontal asymptote by examining the degrees of the numerator and denominator. The degree of the numerator (3) is higher than the degree of the denominator (2). This means there is no horizontal asymptote.

Graphing the Function

The simplified expression and analysis of the zeros and asymptotes provide enough information to start graphing the function.

The graph will cross the x-axis at x = 3.
There will be no vertical asymptotes.
The graph will grow without bound as x approaches positive or negative infinity.

The graph will have a more complex shape than a simple linear or quadratic function, but the information we've gathered helps us understand its key features.