In Exercises 25–32, find the zeros for each 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)=2(x−5)(x+4)^2

Verified step by step guidance

1. Identify the zeros of the polynomial by setting each factor equal to zero: \(2(x-5)(x+4)^2 = 0\).
2. Solve each equation: \(x-5=0\) gives \(x=5\) and \((x+4)^2=0\) gives \(x=-4\).
3. Determine the multiplicity of each zero: \(x=5\) has a multiplicity of 1, and \(x=-4\) has a multiplicity of 2.
4. Analyze the behavior of the graph at each zero: A zero with an odd multiplicity (like \(x=5\)) means the graph crosses the x-axis. A zero with an even multiplicity (like \(x=-4\)) means the graph touches the x-axis and turns around.
5. Summarize: The zero \(x=5\) has multiplicity 1 and the graph crosses the x-axis. The zero \(x=-4\) has multiplicity 2 and the graph touches the x-axis and turns around.

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial function are the values of x for which the function equals zero. These points are where the graph intersects the x-axis. To find the zeros, we set the polynomial equal to zero and solve for x, which often involves factoring the polynomial.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular zero appears as a factor in the polynomial. If a zero has an even multiplicity, the graph touches the x-axis and turns around at that point. Conversely, if the multiplicity is odd, the graph crosses the x-axis at that zero.

Graph Behavior at Zeros

The behavior of a polynomial graph at its zeros is determined by the multiplicity of each zero. For zeros with odd multiplicity, the graph will cross the x-axis, while for zeros with even multiplicity, it will touch the x-axis and turn around. Understanding this behavior helps in sketching the graph of the polynomial function.