1. The problem asks us to use the Intermediate Value Theorem (IVT) to show that the polynomial $$f(x) = 4x^4 - 7x^2 + 2$$ has a real zero between $$-1$$ and $$0$$. 2. The Intermediate Value Theorem states that if a function $$f$$ is continuous on a closed interval $$[a,b]$$ and $$f(a)$$ and $$f(b)$$ have opposite signs, then there exists at least one $$c$$ in $$[a,b]$$ such that $$f(c) = 0$$. 3. First, evaluate $$f(-1)$$: $$f(-1) = 4(-1)^4 - 7(-1)^2 + 2 = 4(1) - 7(1) + 2 = 4 - 7 + 2 = -1$$ 4. Next, evaluate $$f(0)$$: $$f(0) = 4(0)^4 - 7(0)^2 + 2 = 0 - 0 + 2 = 2$$ 5. Since $$f(-1) = -1 < 0$$ and $$f(0) = 2 > 0$$, the function changes sign over the interval $$[-1,0]$$. 6. Because $$f(x)$$ is a polynomial, it is continuous everywhere, including on $$[-1,0]$$. 7. By the Intermediate Value Theorem, there must be some $$c$$ in $$[-1,0]$$ such that $$f(c) = 0$$. Therefore, the polynomial $$4x^4 - 7x^2 + 2$$ has at least one real zero between $$-1$$ and $$0$$.