# Chapter 4 - Polynomial and Rational Functions - 4.5 The Real Zeros of a Polynomial Function - 4.5 Assess Your Understanding - Page 234: 80

$f(-1)=-6$ $f(0)=2$ Since $-6\lt0\lt2$, according to the Intermediate Value Theorem, the function must have a $0$ in the given interval.

#### Work Step by Step

The Intermediate Value Theorem says that if a continuous (polynomials are always continuous) function on an interval $[a,b]$ takes values $f(a)$ and $f(b)$ at the endpoints, then the function takes all values between $f(a)$ and $f(b)$ at some point of the interval. Evaluate the function at the endpoints. $f(-1)=(-1)^4+8\cdot(-1)^3-(-1)^2+2=1-8+1+2=-6$ $f(0)=0^4+8\cdot0^3-0^2+2=0+0-0+2=2$ Since $-6\lt0\lt2$, according to the Intermediate Value Theorem, the function must have a $0$ in the given interval.

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