document.write( "Question 350349: Given P(x) = x^3 − 2x^2 + 9x − 18 \r
\n" ); document.write( "\n" ); document.write( "Factor P(x) completely into linear factors with complex coefficients. \n" ); document.write( "

Algebra.Com's Answer #250481 by jsmallt9(3758) $\"\"$ $\"About$

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$\"P%28x%29+=+x%5E3+-+2x%5E2+%2B+9x+-+18\"$
\n" ); document.write( "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.

\n" ); document.write( "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:
\n" ); document.write( " $\"P%28x%29+=+x%5E2%2Ared%28%28x+-+2%29%29+%2B+9%2Ared%28%28x+-+2%29%29\"$
\n" ); document.write( "As you can see, in red, the two groups have a common factor: (x-2). Factoring this common factor from each group we get:
\n" ); document.write( " $\"P%28x%29+=+red%28%28x-2%29%29%28x%5E2+%2B+9%29\"$
\n" ); document.write( "Since $\"%28x%5E2+%2B+9%29\"$ 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:
\n" ); document.write( "Rewrite as a subtraction:
\n" ); document.write( " $\"x%5E2+-+%28-9%29%29\"$
\n" ); document.write( "Think of this as a difference of squares. It's easy to think of $\"x%5E2\"$ 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.)
\n" ); document.write( "Now that we know what to square to get -9 we and look at $\"x%5E2+-+%28-9%29%29\"$ as a difference of squares and use the pattern of the same name, $\"a%5E2+-+b%5E2+=+%28a+%2B+b%29%28a+-+b%29\"$ , to factor $\"x%5E2+-+%28-9%29%29\"$ into $\"%28x+%2B3i%29%28x-3i%29\"$ . So now we have:
\n" ); document.write( " $\"P%28x%29+=+%28x-2%29%28x%2B3i%29%28x-3i%29\"$
\n" ); document.write( "We have have P(x) factored into linear factors. \n" ); document.write( "

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