Question

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

Accepted Answer

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.

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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Key Concepts

Polynomial Functions and Their Zeros

Intermediate Value Theorem

Bounding Zeros of Polynomials