Question

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 points. So, the number of real zeros is 5.

Step2: Determine sign of polynomial

As \(x\to+\infty\), \(y\to+\infty\). For a polynomial \(y = a_nx^n+\cdots+a_0\), when \(n\) is even and \(a_n> 0\) or \(n\) is odd and \(a_n>0\), \(y\to+\infty\) as \(x\to+\infty\). Since the graph has an odd - number of turning points, the degree is odd and the leading coefficient is positive. So the polynomial is positive.

Step3: Determine degree parity

The number of turning points of a polynomial function \(y = f(x)\) is at most \(n - 1\), where \(n\) is the degree of the polynomial. The graph has 4 turning points, so the degree \(n\) is odd (\(n=4 + 1\) or higher odd number).

Step4: Analyze end - behavior

As \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\) because for an odd - degree polynomial with a positive leading coefficient, the end - behavior is \(y\to-\infty\) as \(x\to-\infty\) and \(y\to+\infty\) as \(x\to+\infty\).

Answer:

• How many real zeros (x - intercepts) does it have? 5
• Is the polynomial positive or negative? Positive
• Is this an odd or even degree polynomial? Odd
• What is the end - behavior? As \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\)