Question

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|

Answer 1

`\begin{gathered} x\text{ = 0 (multiplicity of 2)} \\ x\text{ = }\sqrt[]{7}\text{ (multiplicity of 1)} \\ x\text{ = - }\sqrt[]{7}\text{ (multiplicity of 1)} \end{gathered}`

Step-by-step explanation

`\begin{gathered} f(x)=-7x^2(x^2-7) \\ \text{set }-7x^2(x^2-7)\text{ = 0} \\ x^2\text{ = 0} \\ x\text{ = +-}\sqrt[]{0} \\ x\text{ = 0}\ldots\ldots..\ldots..\ldots\ldots.(1) \\ x^2\text{ - 7 = 0} \\ x^2\text{ = 7} \\ x\text{ = +-}\sqrt[]{7} \\ x\text{ = }\sqrt[]{7}\text{ or x = -}\sqrt[]{7}\ldots\ldots\ldots....\ldots\ldots\text{....}\mathrm{}.(2) \end{gathered}`

The final solution is all the values that make -7x^2(x^2 -7) = 0 true.

while the multiplicity of a root is the number of times the root appears.

for the multiplicity:

`\begin{gathered} x\text{ = 0 (multiplicity of 2)} \\ x\text{ = }\sqrt[]{7}\text{ (multiplicity of 1)} \\ x\text{ = - }\sqrt[]{7}\text{ (multiplicity of 1)} \end{gathered}`