Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x4+2x3−4x2+x−1ƒ(x)=$$3x^4+2x^3$-4x^2+x-1$$; no real zero less than -2

Accepted Answer

First, understand the problem: we need to show that the real zeros of the polynomial function $$f(x) = 3x^4$ + 2x^3$ - 4x^2$ + x - 1$$ are all greater than or equal to $$-2$$, meaning there are no real zeros less than $$-2. $$Evaluate the polynomial at $$x = -2 to $$check the sign of $$f(-2). $$Substitute $$x = -2$$ into the polynomial: $$f(-2) = 3(-2)^4$ + 2(-2)^3$ - 4(-2)^2$ + (-2) - 1. $$Next, analyze the behavior of $$f(x)$$ for values less than $$-2. $$One way is to check the sign of $$f(x) at a $$value less than $$-2$$, for example at $$x = -3$$, by substituting $$x = -3$$ into the polynomial. If $$f(-2)$$ and $$f(-3)$$ have the same sign (both positive or both negative), then by the Intermediate Value Theorem, there is no zero between $$-3$$ and $$-2. $$Repeat this for other values less than $$-2 if $$necessary to confirm no sign changes occur. Finally, conclude that since the polynomial does not change sign for $$x \u003c -2$$, there are no real zeros less than $$-2. $$This shows all real zeros satisfy the condition of being greater than or equal to $$-2.$$

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
$ƒ (x) = 3 x^{4} + 2 x^{3} - 4 x^{2} + x - 1$ ; no real zero less than -2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Real Zeros of Polynomial Functions

Evaluating Polynomial Values to Test Inequalities

Intermediate Value Theorem

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x4+2x3−4x2+x−1ƒ(x)=3x^4+2x^3-4x^2+x-1; no real zero less than -2