Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.ƒ(x)=x5+2x3−2x2+5x+5ƒ(x)=$$x^5+2x^3$-2x^2+5x+5$$; no real zero less than -1

Accepted Answer

First, understand the problem: we need to show that the polynomial function $$f(x) = x^5$ + 2x^3$ - 2x^2$ + 5x + 5$$ has no real zeros less than $$-1. $$This means that for all $$x \u003c -1$$, $$f(x) 
eq 0. $$Step 1: Evaluate the function at $$x = -1 to $$get a reference value. Calculate $$f(-1) = (-1)^5$ + 2(-1)^3$ - 2(-1)^2$ + 5(-1) + 5. $$Step 2: Analyze the behavior of $$f(x)$$ for values less than $$-1. $$Consider the sign of $$f(x)$$ for $$x \u003c -1 by $$testing a value less than $$-1$$, for example $$x = -2$$, and calculate $$f(-2). $$Step 3: Use the Intermediate Value Theorem and the sign test results to argue whether $$f(x)$$ can cross zero for $$x \u003c -1. If$$ $$f(x)$$ does not change sign in that interval, then there are no zeros less than $$-1. $$Step 4: Optionally, analyze the derivative $$f'(x) to $$understand the increasing or decreasing behavior of $$f(x)$$ for $$x \u003c -1$$, which can help confirm the absence of zeros in that region.

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
$ƒ (x) = x^{5} + 2 x^{3} - 2 x^{2} + 5 x + 5$ ; no real zero less than -1

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Key Concepts

Real Zeros of Polynomial Functions

Polynomial Inequalities and Root Bounds

Evaluating Polynomial Values and Sign Changes

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.ƒ(x)=x5+2x3−2x2+5x+5ƒ(x)=x^5+2x^3-2x^2+5x+5; no real zero less than -1