Question

consider the following sketch of a polynomial. what is the end - behavior? as x→ - ∞, y→ < and as x→ + ∞, y→ < is this an odd or even degree polynomial? < is the polynomial positive or negative? < how many real zeros (x - intercepts) does it have? <

Explanation:

Step1: Analyze end - behavior

As \(x\to-\infty\), \(y\to +\infty\) and as \(x\to+\infty\), \(y\to-\infty\). This indicates a negative - leading - coefficient polynomial of odd degree. For an odd - degree polynomial \(y = a_nx^n+\cdots+a_0\) (\(n\) odd), if \(a_n<0\), the end - behavior is \(y\to+\infty\) as \(x\to-\infty\) and \(y\to-\infty\) as \(x\to+\infty\).

Step2: Determine number of real zeros

The graph crosses the \(x\) - axis at 3 points. The real zeros of a polynomial are the \(x\) - values where \(y = 0\), which are the \(x\) - intercepts. So, the number of real zeros is 3.

Answer:

1. As \(x\to-\infty\), \(y\to +\infty\); as \(x\to+\infty\), \(y\to-\infty\)
2. Odd degree
3. Negative
4. 3