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?

1 Answer
George C.
Aug 31, 2015

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:

enter image source here

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

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