Question

Factor the expression: x^3 - 12x^2 + 45x - 69 = 0

Factor the expression:

x^3 - 12x^2 + 45x - 69 = 0

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Homework Answers

Answer #1

It is difficult to factor a cubic polynomial by trial and error. Here, the rational/integral roots theorem also does not help as the only factors of 69 are -3,3,-23 and 23. Also, the only factors of x are -1 and 1. Further, none of -3,3,-23 and 23 is a zero of p(x) = x3 - 12x2 + 45x – 69.

The only other option is to sketch a graph. A graph of p(x) is attached. It may be observed that the graph of p(x) crosses the X-Axis only at x = 6.958 (approximately as may be seen in the original desmos graph). The other 2 zeros of p(x) are, therefore, a pair of conjugate complex numbers. Since it is not possible to locate an exact integral or rational zero of p(x), hence p(x)= x3 - 12x2 + 45x – 69 is irreducible over Z or Q.

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