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Determine the multiplicity of the zeros for each function.

A. f(x)=3x-2

B. f(x)=(x+3)^2(2x-1)

C. f(x)=(x+2)(2x+5)

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Final answer:

A linear function has a single zero with a multiplicity of 1. Quadratic functions can have multiple zeros with varying multiplicities.

Step-by-step explanation:

A. The function f(x) = 3x - 2 is a linear function with a slope of 3. Since it is a linear function, it has a single zero, which is the x-intercept.

To find the x-intercept, set f(x) = 0 and

solve for x: 3x - 2 = 0.

Adding 2 to both sides of the equation gives 3x = 2.

Dividing both sides by 3 gives x = 2/3. So the multiplicity of the zero for this function is 1.

B. The function f(x) = (x + 3)^2(2x - 1) is a quadratic function with two zeros. The factor (x + 3)^2 means that the zero x = -3 has a multiplicity of 2, which means it appears twice.

The factor 2x - 1 has a linear factor, so its zero has a multiplicity of 1.

Therefore, the multiplicity of the zeros for this function are 2 and 1.

C. The function f(x) = (x + 2)(2x + 5) is a quadratic function with two zeros. The factor (x + 2) has a linear factor, so its zero has a multiplicity of 1.

The factor 2x + 5 also has a linear factor, so its zero has a multiplicity of 1 as well.

Therefore, the multiplicity of the zeros for this function are both 1.

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