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: Count real - zeros

The graph crosses the x - axis at 5 distinct points. So the number of real zeros (x - intercepts) is 5.

Step2: Determine sign of polynomial

As x approaches positive infinity, y approaches negative infinity. For a polynomial \(y = a_nx^n+\cdots+a_0\), when \(x\to+\infty\) and \(y\to-\infty\), the leading coefficient \(a_n<0\). So the polynomial is negative.

Step3: Determine degree

Since the graph has 4 turning points, and for a polynomial of degree \(n\), the maximum number of turning points is \(n - 1\). So \(n-1 = 4\), then \(n=5\) (an odd - degree polynomial).

Step4: Analyze end - behavior

As \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to-\infty\) which is consistent with an odd - degree polynomial with a negative leading coefficient.

Answer:

1. 5
2. Negative
3. Odd
4. As \(x\to-\infty\), \(y\to+\infty\); as \(x\to+\infty\), \(y\to-\infty\)