Odd-degree polynomials have at least one x-intercept.
True
False
Answer (2)
Odd-degree polynomials have opposite end behaviors as x approaches positive and negative infinity.
Polynomials are continuous functions, so the Intermediate Value Theorem applies.
By the Intermediate Value Theorem, odd-degree polynomials must cross the x-axis at least once.
Therefore, odd-degree polynomials have at least one x-intercept, and the answer is T r u e .
Explanation
Understanding the Problem An odd-degree polynomial is a polynomial where the highest power of the variable is an odd number (e.g., x 3 + 2 x − 1 ). We need to determine if such polynomials always have at least one x-intercept. An x-intercept occurs where the polynomial's value is zero (i.e., where the graph of the polynomial crosses the x-axis).
Analyzing End Behavior Consider the end behavior of an odd-degree polynomial, p ( x ) . As x approaches positive infinity ( x → ∞ ), p ( x ) will either approach positive infinity ( p ( x ) → ∞ ) or negative infinity ( p ( x ) → − ∞ ). Conversely, as x approaches negative infinity ( x → − ∞ ), p ( x ) will approach the opposite infinity. For example, if p ( x ) → ∞ as x → ∞ , then p ( x ) → − ∞ as x → − ∞ , and vice versa.
Applying the Intermediate Value Theorem Since polynomials are continuous functions (meaning their graphs have no breaks or jumps), the Intermediate Value Theorem applies. This theorem states that if a continuous function has values of opposite signs at two points, it must have at least one root (i.e., an x-intercept) between those points. Because an odd-degree polynomial goes from − ∞ to + ∞ (or vice versa), it must cross the x-axis at least once.
Concluding the Answer Therefore, an odd-degree polynomial must have at least one real root, which corresponds to an x-intercept. This means the statement 'Odd-degree polynomials have at least one x-intercept' is true.
Examples
Consider designing a rollercoaster track. The height of the track can be modeled by a polynomial function. If you need to ensure the track crosses a certain height (the x-axis in a simplified model) at least once, using an odd-degree polynomial guarantees that the track will indeed cross that height. This is because odd-degree polynomials always have at least one real root, ensuring the track intersects the desired height. This concept is crucial in ensuring the rollercoaster design meets specific safety and design criteria.
The statement 'Odd-degree polynomials have at least one x-intercept' is true. Odd-degree polynomials take on opposite end behaviors, ensuring they cross the x-axis at least once due to the Intermediate Value Theorem. Therefore, they must have at least one real root or x-intercept.
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