Question

what is the graph of the function f(x)= x^3+x^2+x+1

Answer 1

In order to sketch the graph of this function, it is useful to gather the following information:

• $x=-1$ is a solution for this equation.
• If you long divide $(x^3+x^2+x+1)\div(x+1)$ you get $x^2+1$. This means that we can write $x^3+x^2+x+1 = (x+1)(x^2+1)$
• Since $x^2+1$ has no roots, it means that $x=-1$ is the only root of this polynomial
• This is a cubic polynomial, and the leading term has a positive coefficient. This means that $\lim_{x\to\pm\infty}f(x)=\pm\infty$

So, you have to draw a cubic polynomial crossing the x axis at $x=-1$, which is negative before the root and positive afterwards, and tends to negative/positive infinity as x approaches negative/positive infinity.