We are given y=x^3+4x^2-x-4y=x3+4x2−x−4. Because the highest exponent is 33, we know there are 33 solutions.
With anything larger than a quadratic (ax^2+bx+cax2+bx+c), I like to use synthetic division. To use synthetic division, the first thing we must do is find the divisor. We can do that by guessing-and-checking, or we can graph the equation and use the roots (x-intercepts) as our divisor. Let's do that right now:
graph{y=x^3+4x^2-x-4}
Lucky for us, we can see all three solutions. I'm still going to use synthetic division just to show you how it works:
color(white)(-1.)|−1.∣color(white)(.)1.1color(white)(...)4color(white)(.)-1color(white)(.)-4
-1color(white)(.)|_-1__-3__4_____
color(white)(.........)1color(white)(...)3color(white)(..)-4color(white)(....)0
This can be rewritten as x^2+3x-4, which can be factored to (x-1)(x+4).
So, we have (x+4)(x-1)(x+1)! And that's what the graph shows too. Nice work!
