Question

1. the function g(x) approaches positive infinity as x approaches positive infinity. the zeros of the function are 1, -2, and -3. graph g(x). write an equation for g(x)

Explanation:

Step1: Identify the roots and form factors

The zeros of the function are \( x = 1 \), \( x=-2 \), and \( x = -3 \). So the factors of the polynomial are \( (x - 1) \), \( (x + 2) \), and \( (x + 3) \).

Step2: Determine the leading coefficient

Since \( g(x) \) approaches \( +\infty \) as \( x \to +\infty \), the leading coefficient must be positive. Let's assume the leading coefficient is \( a = 1 \) (we can choose any positive real number, 1 is simple).

Step3: Write the polynomial equation

Multiply the factors together:

$$ g(x)=(x - 1)(x + 2)(x + 3) $$

We can expand this to check:
First, multiply \( (x + 2)(x + 3)=x^{2}+5x + 6 \)
Then, multiply by \( (x - 1) \):

$$
LATEXBLOCK0
$$

The leading term is \( x^{3} \), and as \( x\to+\infty \), \( x^{3}\to+\infty \), which satisfies the end - behavior condition.

Answer:

\( g(x)=(x - 1)(x + 2)(x + 3) \) (or the expanded form \( g(x)=x^{3}+4x^{2}+x - 6 \))