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When dealing with polynomial functions, the zeros are the values of \(x\) that make the polynomial equal to zero. To find these zeros, you set the polynomial equation to zero and solve for \(x\). In simpler terms, zeros are the points where the graph of the polynomial intersects the \(x\)-axis. For the exercise involving the polynomial function \(f(x) = 2(x-5)(x+4)^2\), the equation is set to zero as follows:
\[2(x - 5)(x + 4)^2 = 0\]This equation is satisfied when the expressions inside the parentheses are zero, giving the potential zeros:

• \((x-5) = 0\) leading to \(x = 5\)
• \((x + 4)^2 = 0\) leading to \(x = -4\)

Thus, the zeros of this polynomial function are \(x = 5\) and \(x = -4\). These are the points where the polynomial graph will intersect or touch the \(x\)-axis.

Multiplicity is an important concept when discussing the zeros of a polynomial. It refers to how many times a particular solution appears as a root of the polynomial. This is determined by examining the exponent of the factor corresponding to a zero.

For the polynomial \(f(x) = 2(x-5)(x+4)^2\), each factor that gives a zero has a certain exponent:

• The factor \((x-5)\) contributes to the zero \(x = 5\) with an exponent of 1, giving it a multiplicity of 1.
• The factor \((x + 4)^2\) contributes to the zero \(x = -4\) and is raised to the power of 2, thus having a multiplicity of 2.

The multiplicity affects the behavior of the graph at each zero, indicating whether the graph crosses the \(x\)-axis or merely touches it and turns around.

The behavior of a polynomial graph at its zeros is largely determined by the multiplicity of each zero. Understanding this concept helps us predict whether the graph will cross or touch the \(x\)-axis:

• **Odd Multiplicity**: If a zero has an odd multiplicity, the graph will **cross** the \(x\)-axis at that zero.
• **Even Multiplicity**: If a zero has an even multiplicity, the graph will **touch** the \(x\)-axis and turn around at that zero.

Applying this to our polynomial function \(f(x) = 2(x-5)(x+4)^2\):

• At \(x = 5\), where the multiplicity is 1 (odd), the graph crosses the \(x\)-axis.
• At \(x = -4\), where the multiplicity is 2 (even), the graph touches the \(x\)-axis and turns back.

This characteristic behavior at each zero can help us sketch the graph of a polynomial function by marking these important interactions with the \(x\)-axis.