Question

How do you use the graph of `f(x) =x^3-6x^2+11x-6` to rewrite `f(x)` as a product of linear factors?

Answer 1

`f(x)=color(green)((x-1)(x-2)(x-3))`

Explanation:

Here is the graph of `f(x)=x^3-6x^2+11x-6`

Notice that the graph crosses the X-axis at three points.

`f(x)=0` when :
`color(white)("XXX")x=1color(white)("XX")rarrcolor(white)("XX")(x-1)` is a factor of `f(x)`
`color(white)("XXX")x=2color(white)("XX")rarrcolor(white)("XX")(x-2)` is a factor of `f(x)`
`color(white)("XXX")x=3color(white)("XX")rarrcolor(white)("XX")(x-3)` is a factor of `f(x)`

Since `f(x)` is of degree `3` it has a maximum of `3` factors,
so these `3` represent a complete factoring of `f(x)`.