Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=3x⁴+2x³-4x²+x-1; no real zero greater than 1

Verified step by step guidance

1. First, understand the problem: we need to show that the real zeros of the polynomial function \(f(x) = 3x^4 + 2x^3 - 4x^2 + x - 1\) satisfy the condition that no real zero is greater than 1.
2. Evaluate the polynomial at \(x = 1\) to check the sign of \(f(1)\). Substitute \(x = 1\) into the polynomial: \(f(1) = 3(1)^4 + 2(1)^3 - 4(1)^2 + 1 - 1\).
3. Analyze the behavior of \(f(x)\) for values greater than 1. For example, evaluate \(f(2)\) or consider the end behavior of the polynomial to see if it can cross the x-axis beyond \(x=1\).
4. Use the Intermediate Value Theorem: if \(f(1)\) and \(f(x)\) for some \(x > 1\) have the same sign, then there is no zero between 1 and that \(x\). If \(f(x)\) does not change sign for \(x > 1\), then no zeros exist greater than 1.
5. Optionally, find the critical points by differentiating \(f(x)\) and analyze the function's increasing or decreasing behavior to support the conclusion that no real zeros are greater than 1.

Key Concepts

Real Zeros of Polynomial Functions

Real zeros of a polynomial are the values of x for which the polynomial equals zero. These zeros correspond to the x-intercepts of the graph. Understanding how to find and interpret real zeros is essential for analyzing the behavior of polynomial functions.

Evaluating Polynomial Functions at Specific Points

Evaluating a polynomial at a given value involves substituting that value into the function and calculating the result. This helps determine whether a number is a zero or to check the sign of the polynomial at certain points, which is useful for bounding the location of zeros.

Using the Intermediate Value Theorem and Sign Analysis

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, it must have a zero in that interval. By analyzing the sign of the polynomial at points around 1, one can show that no real zero exists greater than 1, confirming the given condition.