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 less than -3

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First, understand that to show a polynomial function has a real zero in a certain interval, 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 (zero) in that interval.

Evaluate the polynomial function at the endpoints of the interval given. Here, since we want to show there is no zero less than -3, start by evaluating \(f(-3)\) and also consider values less than -3, for example \(f(-4)\) or \(f(-5)\).

Calculate \(f(-3) = 6(-3)^4 + 13(-3)^3 - 11(-3)^2 - 3(-3) + 5\) and similarly calculate \(f(-4)\) or \(f(-5)\) by substituting these values into the polynomial. Note the signs of these function values.

Check if the function values at these points change sign. If \(f(-3)\) and \(f(-4)\) (or \(f(-5)\)) have the same sign, then by the Intermediate Value Theorem, there is no zero between these points, meaning no zero less than -3.

Summarize your findings: if the function does not change sign for any values less than -3, then the polynomial has no real zeros less than -3, as required.

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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 and interpret these zeros is essential for analyzing the behavior of the polynomial.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, then it must have at least one zero within that interval. This theorem is useful for proving the existence of real zeros of polynomial functions on specific intervals.

Bounding Zeros of Polynomials

Bounding zeros involves determining intervals where zeros cannot exist, such as showing no zeros are less than a certain value. Techniques include evaluating the polynomial at boundary points and using sign analysis to restrict the location of zeros.