Which of the following is true about the roots of a polynomial of odd degree?

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Question: Which of the following is true about the roots of a polynomial of odd degree?

Options:

Correct Answer: It has at least one real root.

Solution:

A polynomial of odd degree must have at least one real root due to the Intermediate Value Theorem.

Which of the following is true about the roots of a polynomial of odd degree?

Which of the following is true about the roots of a polynomial of odd degree?

Step 1: Understand what a polynomial is. A polynomial is a mathematical expression that can have variables raised to whole number powers, like x^3 + 2x + 1.
Step 2: Identify what is meant by 'odd degree'. The degree of a polynomial is the highest power of the variable. An odd degree means the highest power is an odd number, like 1, 3, 5, etc.
Step 3: Learn about the Intermediate Value Theorem. This theorem states that if a continuous function (like a polynomial) takes on two different values at two points, it must cross every value in between those two points.
Step 4: Apply the Intermediate Value Theorem to odd degree polynomials. Since odd degree polynomials go to positive infinity on one end and negative infinity on the other, they must cross the x-axis at least once.
Step 5: Conclude that because of the Intermediate Value Theorem, a polynomial of odd degree must have at least one real root.

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