Question 350349: Given P(x) = x^3 − 2x^2 + 9x − 18
Factor P(x) completely into linear factors with complex coefficients. Answer by jsmallt9(3758) (Show Source):
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When factoring always start with the Greatest Common Factor (GCF). The GCF of all four terms is 1. Since we rarely factor out a 1, we will move on to other factoring techniques. The most commonly used patterns all have 2 or 3 terms. Since P(x) has 4 terms it looks like patterns won't help. There is also trimomial factoring but, as the name implies, it is for three term expressions.
So we are left with Factoring by grouping or by trial and error of the possible rational roots. For factoring by grouping you look for groups of terms that have a GCF. In P(x) the first two terms have a GCF (other then 1) and the last two terms also have a GCF (other than 1) so this looks promising. Factoring out the GCF of each group we get:
As you can see, in red, the two groups have a common factor: (x-2). Factoring this common factor from each group we get:
Since is not linear we must continue trying to factor. This will not be simple. And I'm not sure how you've been taught to do something like this. But here's one way:
Rewrite as a subtraction:
Think of this as a difference of squares. It's easy to think of as a perfect square. But what to so square to get -9?? If you know about Real and Imaginary numbers you might realize that if yous square 3i (or -3i) you get -9! (If this is not clear, try squaring 3i and see what you get.)
Now that we know what to square to get -9 we and look at as a difference of squares and use the pattern of the same name, , to factor into . So now we have:
We have have P(x) factored into linear factors.
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