Find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. $f\left(x\right)=-2(x-1)(x+2{)}^2(x+5{)}^3$

Verified step by step guidance

1. Identify the zeros of the polynomial by setting each factor equal to zero: solve \(x - 1 = 0\), \(x + 2 = 0\), and \(x + 5 = 0\) to find the zeros.
2. Determine the multiplicity of each zero by looking at the exponent on each factor: the factor \((x - 1)\) has multiplicity 1, \((x + 2)^2\) has multiplicity 2, and \((x + 5)^2\) has multiplicity 2.
3. Recall that if a zero has an odd multiplicity, the graph crosses the x-axis at that zero; if the multiplicity is even, the graph touches the x-axis and turns around at that zero.
4. Summarize the behavior at each zero: for \(x = 1\) (multiplicity 1), the graph crosses the x-axis; for \(x = -2\) and \(x = -5\) (both multiplicity 2), the graph touches and turns around at the x-axis.
5. Combine all this information to describe the zeros, their multiplicities, and the behavior of the graph at each zero.

Key Concepts

Zeros of a Polynomial Function

Zeros of a polynomial are the values of x that make the function equal to zero. They correspond to the x-intercepts of the graph. To find zeros, set the polynomial equal to zero and solve for x, often by factoring or using given factored form.

Multiplicity of Zeros

Multiplicity refers to how many times a particular zero appears as a factor in the polynomial. If a zero has even multiplicity, the graph touches the x-axis and turns around at that zero. If the multiplicity is odd, the graph crosses the x-axis at that zero.

Behavior of Graph at Zeros

The graph's behavior at each zero depends on the zero's multiplicity. For odd multiplicity, the graph crosses the x-axis, indicating a sign change. For even multiplicity, the graph only touches the axis and reverses direction, showing no sign change.