# How do you use the Intermediate Value Theorem to show that the polynomial function #x^3 - 2x^2 + 3x = 5# has a zero in the interval [1, 2]?

Assuming you meant

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Here's one way to do it.

That is, the original equation has a solution.

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To use the Intermediate Value Theorem to show that the polynomial function (x^3 - 2x^2 + 3x - 5) has a zero in the interval ([1, 2]), you need to demonstrate that the function changes sign within that interval. First, evaluate the function at the endpoints of the interval: (f(1)) and (f(2)). If the values have different signs, then by the Intermediate Value Theorem, there exists at least one value (c) in the interval ([1, 2]) such that (f(c) = 0).

(f(1) = (1)^3 - 2(1)^2 + 3(1) - 5 = -3)

(f(2) = (2)^3 - 2(2)^2 + 3(2) - 5 = 1)

Since (f(1) = -3) and (f(2) = 1), and they have different signs, the Intermediate Value Theorem guarantees that there exists at least one value (c) in the interval ([1, 2]) such that (f(c) = 0), which means the polynomial has a zero in the interval ([1, 2]).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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