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What are the possible numbers of positive, negative, and complex zeros of f(x) = x4 + 3x3 − 2x2 + 1? (6 points)

Positive: 2 or 0; negative: 2 or 0; complex: 4, 2, or 0

Positive: 2, or 0; negative: 4, 2 or 0; complex: 4, 2, or 0

Positive: 2 or 0; negative: 2 or 0; complex: 4, 2, or 0

Positive: 2 or 0; negative: 0; complex: 2, or 0

real: 4,2,0

Using Descartes' Rule of Signs,
positive: 2 max
negative: 2 max
complex: 4,2,0

how come (a),(b),(c) are all the same?

To determine the possible numbers of positive, negative, and complex zeros of a polynomial function, you can use Descartes' Rule of Signs.

Here's how you can apply Descartes' Rule of Signs to find the number of positive zeros:

1. Count the number of sign changes in the coefficients of the terms when you rewrite the function in standard form. In this case, the polynomial is already in standard form: f(x) = x^4 + 3x^3 - 2x^2 + 1.

The sign changes in coefficients occur from +1 to +3 and from +3 to -2. So there are 2 sign changes.

2. The number of positive zeros is either equal to the number of sign changes or less than it by an even integer.

Since there are 2 sign changes, the possible numbers of positive zeros are 2 or 0.

To find the possible numbers of negative zeros, follow the same process but now consider the function f(-x) instead of f(x). In this case, f(-x) = (-x)^4 + 3(-x)^3 - 2(-x)^2 + 1.

The sign changes in coefficients occur from +1 to +3 and from +3 to +2. So there are 2 sign changes.

Again, the number of negative zeros is either equal to the number of sign changes or less than it by an even integer. Thus, the possible numbers of negative zeros are 2 or 0.

Now, let's move on to complex zeros. For a polynomial with real coefficients, complex zeros occur in conjugate pairs. Since the polynomial f(x) = x^4 + 3x^3 - 2x^2 + 1 has degree 4, there can be either 0, 2, or 4 complex zeros.

In summary, the possible numbers of positive, negative, and complex zeros of the given function f(x) = x^4 + 3x^3 - 2x^2 + 1 are:
- Positive zeros: 2 or 0
- Negative zeros: 2 or 0
- Complex zeros: 4, 2, or 0