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.
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. For example, x 3 + 2 x − 1 is an odd-degree polynomial. The question asks whether such polynomials always have at least one x-intercept. An x-intercept is a point where the polynomial's value is zero, meaning the graph of the polynomial crosses the x-axis at that point.
Analyzing End Behavior To determine whether the statement is true or false, we need to consider the end behavior of odd-degree polynomials. The end behavior describes what happens to the polynomial's value as x approaches positive infinity ( x → ∞ ) and negative infinity ( x → − ∞ ).
Describing End Behavior For an odd-degree polynomial, as x approaches positive infinity, the polynomial will approach either positive infinity or negative infinity. Conversely, as x approaches negative infinity, the polynomial will approach the opposite infinity from its behavior at positive infinity. For example, if the polynomial approaches positive infinity as x approaches positive infinity, it will approach negative infinity as x approaches negative infinity, and vice versa.
Applying the Intermediate Value Theorem Polynomials are continuous functions, meaning their graphs have no breaks or jumps. Because of this continuity and the end behavior described above, an odd-degree polynomial must cross the x-axis at least once. This is a consequence of the Intermediate Value Theorem, which states that if a continuous function has values of opposite signs at two different points, it must have a root (i.e., a point where the function equals zero) between those points.
Concluding the Answer Since an odd-degree polynomial must have at least one real root, this root corresponds to an x-intercept. Therefore, the statement 'Odd-degree polynomials have at least one x-intercept' is true.
Examples
Odd-degree polynomials are used in various fields, such as physics and engineering, to model phenomena that exhibit changes in direction or behavior. For example, the trajectory of a projectile under certain conditions can be modeled using an odd-degree polynomial. The x-intercepts of the polynomial would then represent points where the projectile hits the ground or reaches a certain level. Understanding the properties of these polynomials, such as the existence of at least one x-intercept, is crucial for making accurate predictions and analyses in these applications.
Odd-degree polynomials have at least one x-intercept due to their continuity and opposite end behaviors as x approaches positive and negative infinity. This ensures that they must cross the x-axis at least once. Therefore, the answer is True .
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