1. **State the problem:** We want to use the Intermediate Value Theorem (IVT) to show that the polynomial $$f(x) = 4x^4 - 9x^2 + 1$$ has a real zero between $$-1$$ and $$0$$. 2. **Recall the Intermediate Value Theorem:** 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. **Check continuity:** Since $$f(x)$$ is a polynomial, it is continuous everywhere, including on $$[-1,0]$$. 4. **Evaluate $$f$$ at the endpoints:** - $$f(-1) = 4(-1)^4 - 9(-1)^2 + 1 = 4(1) - 9(1) + 1 = 4 - 9 + 1 = -4$$ - $$f(0) = 4(0)^4 - 9(0)^2 + 1 = 0 - 0 + 1 = 1$$ 5. **Check signs:** $$f(-1) = -4 < 0$$ and $$f(0) = 1 > 0$$, so $$f(-1)$$ and $$f(0)$$ have opposite signs. 6. **Apply IVT:** Since $$f$$ is continuous on $$[-1,0]$$ and changes sign from negative to positive, there exists some $$c$$ in $$(-1,0)$$ such that $$f(c) = 0$$. **Final answer:** By the Intermediate Value Theorem, the polynomial $$f(x) = 4x^4 - 9x^2 + 1$$ has at least one real zero between $$-1$$ and $$0$$.