Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.
f(x)=x^{3}+4 x^{2}-9 x-36
Answer
\(\boxed{\text{The zeros of the function are -4, -3, and 3. The multiplicities of these zeros are 7, -6, and 42 respectively. The graph crosses the x-axis at x=-4 and touches the x-axis and turns around at x=-3 and x=3.}}\)
Step 1 ：Set the function equal to zero: \(x^{3}+4 x^{2}-9 x-36 = 0\)
Step 2 ：Solve for x to find the zeros of the function: \(x = -4, -3, 3\)
Step 3 ：Determine the multiplicity of each zero: The multiplicities are 7, -6, and 42 respectively
Step 4 ：Determine whether the graph crosses or touches the x-axis at each zero: The graph crosses the x-axis at x=-4 and touches the x-axis and turns around at x=-3 and x=3
Step 5 ：\(\boxed{\text{The zeros of the function are -4, -3, and 3. The multiplicities of these zeros are 7, -6, and 42 respectively. The graph crosses the x-axis at x=-4 and touches the x-axis and turns around at x=-3 and x=3.}}\)
