Use the constant sign theorem to show that the polynomial y=x^5-4 x^2-x+3 has a root in the inte

step 1 Here we have to show that given polynomial has a 0 in this given interval by intermediate value

Use the constant sign theorem to show that the polynomial...

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Key Concepts

Intermediate Value Theorem

This fundamental theorem in calculus states that if a function is continuous on a closed interval [a, b] and takes on values of opposite signs at the endpoints, then it must cross the x-axis somewhere in between. This property guarantees the existence of at least one real root in the interval when the function values at the endpoints are of opposite signs.

Continuity of Polynomials

Polynomials are continuous functions over the entire set of real numbers. This means that for any interval, a polynomial does not have any breaks, jumps, or holes. This property is crucial for applying the Intermediate Value Theorem since continuity is a required condition for the theorem to hold.

Sign Analysis and Endpoint Evaluation

Evaluating the function at the endpoints of an interval and checking for a sign change is a common method to apply the Intermediate Value Theorem. When a function has one value positive and the other negative at the ends of the interval, it indicates that the function must cross zero at least once within the interval, thereby proving the existence of a root.

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