Question

How do you use a graph, synthetic division, and factoring to find all the roots of an equation `x^3 + 5x^2 + 3x -9 = 0`?

Answer 1

Graphing shows us roots `x=1`, `x=-3` (repeated).

Alternatively, synthetic division and factoring gives us the same roots.

Explanation:

Let `f(x) = x^3+5x^2+3x-9`

The word 'graph' put me off answering this sooner. To plot a graph I usually try to identify turning points, roots, etc first - not the other way round. But if you do plot a graph by picking a few `x` values, then you would pretty quickly find that `x=1` is a root and you would probably notice that `x=-3` is a repeated root.

graph{x^3+5x^2+3x-9 [-10.51, 9.49, -9.56, 0.44]}

Alternatively, first note that the sum of the coefficients is `0`, implying that `x=1` is a root.

So `(x-1)` is a factor. Divide by this using synthetic division:

So `x^3+5x^2+3x-9 = (x-1)(x^2+6x+9)`

`x^2+6x+9` is a perfect square trinomial, recognisable by being of the form `A^2+2AB+B^2 = (A+B)^2` :

`x^2+6x+9 = (x^2+2*x*3+3^2) = (x+3)^2`

Another little trick for recognising this perfect square trinomial is that `169 = 13^2` is a perfect square. So the pattern of coefficients `1`, `6`, `9` corresponds to a square of a binomial with coefficients `1`, `3`.

So `f(x) = (x-1)(x+3)^2` and `f(x) = 0` has roots `1, -3, -3`