Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)^4(x-3), the number 2 is a zero of multiplicity 4.

Verified step by step guidance

1. Identify the given function, which is ƒ(x) = (x+2)^4(x-3).
2. Recognize that the zeros of the function are the values of x that make the function equal to zero. These are found by setting each factor in the function equal to zero.
3. Solve (x+2) = 0 to find one zero. Subtract 2 from both sides to isolate x, giving x = -2.
4. Note that the factor (x+2) is raised to the power of 4, indicating that -2 is a zero of multiplicity 4.
5. Observe that the zero 2 does not appear in the factorization of the function, and thus, the statement 'the number 2 is a zero of multiplicity 4' is false because 2 is not a zero of the function at all.

Key Concepts

Zeros of a Function

A zero of a function is a value of x that makes the function equal to zero. For the function ƒ(x)=(x+2)^4(x-3), the zeros are found by setting the function equal to zero and solving for x. The zero at x = -2 occurs when the factor (x + 2) is equal to zero, indicating that the function crosses the x-axis at this point.

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular zero appears as a factor in the polynomial. In the case of ƒ(x)=(x+2)^4(x-3), the zero at x = -2 has a multiplicity of 4 because the factor (x + 2) is raised to the fourth power. This means that the graph of the function touches the x-axis at x = -2 but does not cross it.

Polynomial Behavior at Zeros

The behavior of a polynomial at its zeros is influenced by the multiplicity of those zeros. If a zero has an odd multiplicity, the graph crosses the x-axis at that zero, while an even multiplicity means the graph touches the x-axis and turns around. For the zero at x = -2 with multiplicity 4, the function will touch the x-axis and remain above or below it, confirming that the statement about the zero is true.