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)=6x^4+13x^3-11x^2-3x+5 no zero greater than 1

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First, understand that the problem asks to show that the polynomial function $f(x) = 6x^4 + 13x^3 - 11x^2 - 3x + 5$ has a real zero, and specifically that there is no zero greater than 1.

Evaluate the polynomial at $x = 1$ by substituting 1 into the function: calculate $f(1) = 6(1)^4 + 13(1)^3 - 11(1)^2 - 3(1) + 5$ to determine the sign of the function at 1.

Evaluate the polynomial at another value less than 1, for example at $x = 0$, by calculating $f(0) = 6(0)^4 + 13(0)^3 - 11(0)^2 - 3(0) + 5$ to check the sign of the function at 0.

Use the Intermediate Value Theorem, which states that if a continuous function changes sign over an interval, then it must have at least one zero in that interval. Since polynomials are continuous everywhere, if $f(0)$ and $f(1)$ have opposite signs, there is at least one zero between 0 and 1.

To show there is no zero greater than 1, analyze the behavior of $f(x)$ for $x > 1$. You can check the sign of $f(2)$ or consider the end behavior of the polynomial (leading term \$6x^4$ dominates for large $x$), which tends to positive infinity, and verify that $f(x)$ does not cross zero for $x > 1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 polynomial.

Bounds on Zeros of Polynomials

Bounds on zeros help determine intervals where the real zeros of a polynomial can lie. Techniques like the Upper and Lower Bound Theorems or the use of synthetic division can show that no zeros exist beyond certain values, such as proving no zero is greater than 1.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - c). It is useful for testing possible zeros and verifying bounds by checking the sign and remainder, which helps confirm whether a given number is a zero or an upper/lower bound.