Question

I am trying to familiarize myself with polynomials but my schooling does an extremely poor job of explaining it. I don't just want a answer I would like for someone to walk through this problem with me.
Graph y=(x + 2)(x + 1)(x –3)^2

Answer 1

now, this polynomial is  y = (x+2)(x+1)(x-3)².

$\bf y=(x+2)(x+1)(x-3)^2\implies 0=(x+2)(x+1)(x-3)(x-3) \\\\\\ \begin{cases} 0=x+2\implies &-2=x\\ 0=x+1\implies &-1=x\\ 0=x-3\implies &~~3=x\\ 0=x-3\implies &~~3=x \end{cases}$

so it has roots there, at those locations, 4 roots, at -2, -1 and 3, but notice the root of 3, is there twice, namely, it has a multiplicity of 2.

quick note on multiplicity:

if the multiplicity is an EVEN value, like 2,4 12, the graph doesn't pass the axis, in this case the x-axis, it simply bounces off of it, and goes right back.

if the multiplicity is an ODD value, the graph does pass the axis.

so, in this case, the root of 3 has a multiplicity of 2, that means, at that point, the graph doesn't cross the axis, it simply bounces off of it.

now, let's get the original polynomial, by simply multiplying the factors,

$\bf y=(x+2)(x+1)(x-3)^2\implies y=(x^2+3x+2)~~~~(x-3)^2 \\\\\\ y=(x^2+3x+2)~~~~(x^2-6x+9) \\\\\\ y=x^4-6x^3+9x^2+3x^3-18x^2+27x+2x^2-12x+18 \\\\\\ y=x^4-3x^3-7x^2+15x+18$

one quick note, when you have polynomials multiplication, you don't have to necessarily do FOIL, you can simply multiply each term by the others at a time, like say

(x+y+z) (t+u+w) => x(t+u+w) + y(t+u+w) + z(t+u+w)

xt+xu+xw + yt+yu+yw + zt+zu+zw

and then combine like terms and simplify.

anyhow, for the graph of this polynomial, check the picture below.