Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). ƒ(x)=4x³-37x²+50x+60 between 7 and 8

Verified step by step guidance

1. First, understand that to show a polynomial function has a real zero between two values, we can use the Intermediate Value Theorem. This theorem states that if a continuous function changes sign over an interval, then it must have at least one root in that interval.
2. Evaluate the function \( f(x) = 4x^3 - 37x^2 + 50x + 60 \) at the endpoints of the interval, which are \( x = 7 \) and \( x = 8 \). Calculate \( f(7) \) and \( f(8) \) separately.
3. Check the signs of \( f(7) \) and \( f(8) \). If one is positive and the other is negative, then by the Intermediate Value Theorem, there is at least one real zero between 7 and 8.
4. To further confirm, you can use methods such as the bisection method or graphing to approximate the root more precisely within the interval.
5. Summarize that since the function is a polynomial (which is continuous everywhere) and the function values at 7 and 8 have opposite signs, the function must have a real zero between 7 and 8.

Key Concepts

Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x for which the function equals zero. Understanding how to find or estimate these zeros is essential for analyzing the behavior of the function.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval [a, b], then it must have at least one root in that interval. This theorem is used to show the existence of a real zero between two points by evaluating the function at those points.

Evaluating Polynomial Functions at Specific Points

To apply the Intermediate Value Theorem, you evaluate the polynomial at given points to check the sign of the function values. If the function values at two points have opposite signs, it confirms a zero exists between those points. This process helps locate intervals containing real zeros.