How can I use the Intermediate Value Theorem to show that the equation
has at least two real solutions.. Thanks.
Looking at the plot in WolframAlpha, it looks like the value at -0.5 is negative. Prove it and use the fact that the polynomial becomes very big and positive when x grows in absolute value. You can easily find values of x for which the polynomial is positive.
Let the polynomial be denoted by f(x). Prove that f(-0.5) < 0 by computing f(-0.5). Further, when . (Since you are studying the Intermediate Value Theorem, which uses continuous functions, you are supposed to know what a limit is.) In fact, it is not necessary to invoke limits; it is easy to find an and an such that and . Thus, f(x) changes sign from to -0.5 and from -0.5 to .