Questions: For the polynomial function below: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of x. f(x)=(x-4)^3(x+5)^2

Solution Steps

The polynomial function is given by \( f(x) = (x - 4)^3 (x + 5)^2 \). To find the real zeros, we set each factor equal to zero:

• \( x - 4 = 0 \) gives \( x = 4 \) with multiplicity 3.
• \( x + 5 = 0 \) gives \( x = -5 \) with multiplicity 2.

Thus, the real zeros and their multiplicities are:

• \( x = 4 \) with multiplicity 3
• \( x = -5 \) with multiplicity 2

The behavior of the graph at each \( x \)-intercept depends on the multiplicity of the zero:

• If the multiplicity is odd, the graph crosses the \( x \)-axis.
• If the multiplicity is even, the graph touches the \( x \)-axis.

For \( x = 4 \) (multiplicity 3), the graph crosses the \( x \)-axis. For \( x = -5 \) (multiplicity 2), the graph touches the \( x \)-axis.

The degree of the polynomial is the sum of the multiplicities, which is \( 3 + 2 = 5 \). The maximum number of turning points is one less than the degree of the polynomial:

\[ \text{Maximum number of turning points} = 5 - 1 = 4 \]

The end behavior of the polynomial is determined by the leading term of the expanded polynomial. The leading term is \( x^5 \), which indicates that for large values of \(|x|\), the graph of \( f(x) \) resembles the power function \( x^5 \).

Final Answer