Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. It is possible for a polynomial with an even degree to have a range of \((-\infty, \infty)\)

Step-by-Step Solution

Answer

The statement is false. An even degree polynomial cannot have a range of all real numbers.

Step 1: Understanding Polynomial Properties

A polynomial function of even degree will either open upwards on both ends if the leading coefficient is positive, or downwards on both ends if the leading coefficient is negative. This is because for even-degree functions, as \(x\) tends toward positive or negative infinity, \(f(x)\) also tends toward positive or negative infinity depending on the sign of the leading coefficient.

Step 2: Assessing Range

From the above understanding, we can infer that an even-degree polynomial will not cover the entire set of real numbers, i.e., it will not have a range of \((-\infty, \infty)\). If the leading coefficient is positive, the range of function will be \([k, \infty)\) where \(k\) is the minimum value of function. If leading coefficient is negative, the range will be \((-\infty, k]\) where \(k\) is the maximum value of the function.

Step 3: Conclusion

From these assessments, the statement that a polynomial with an even degree can have a range of \((-\infty, \infty)\) is FALSE. An even-degree polynomial function will either have a minimum value and extend to positive infinity or have a maximum value and extend to negative infinity, but it cannot extend in both directions on the y-axis.