Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=2x5-x4+2x3-2x2+4x-4; no real zero greater than 1

Accepted Answer

First, understand the problem: we need to show that the polynomial function $$f(x) = 2x^5$ - x^4$ + 2x^3$ - 2x^2$ + 4x - 4$$ has no real zeros greater than 1. This means if $$f(c) = 0$$ for some real number $$c$$, then $$c \leq 1. $$Evaluate the polynomial at $$x = 1 to $$check the sign of $$f(1). $$Substitute $$x=1$$ into the polynomial: $$f(1) = 2(1)^5$ - (1)^4$ + 2(1)^3$ - 2(1)^2$ + 4(1) - 4. $$Analyze the behavior of $$f(x)$$ for values greater than 1. One way is to check the sign of $$f(x) at a $$value greater than 1, for example at $$x=2$$, to see if the polynomial changes sign, which would indicate a zero in that interval. Use the Intermediate Value Theorem: if $$f(1)$$ and $$f(2)$$ have the same sign, then there is no zero between 1 and 2. Repeat this for other values greater than 1 if necessary to confirm no zeros exist beyond 1. Alternatively, consider the polynomial's end behavior and use techniques such as synthetic division or the Rational Root Theorem to test possible roots greater than 1, or analyze the derivative to understand the function's increasing/decreasing behavior beyond $$x=1.$$

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=2x⁵-x⁴+2x³-2x²+4x-4; no real zero greater than 1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Real Zeros of Polynomial Functions

Polynomial Inequalities and Root Bounds

Evaluating Polynomials and Sign Testing