document.write( "Question 824327: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros.
\n" ); document.write( "-2, 2 - i
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Algebra.Com's Answer #496311 by jsmallt9(3758)\"\" \"About 
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A polynomial of degree 3 should have 3 zeros. But only two have been given. The third zero must be found in order to find the polynomial.

\n" ); document.write( "The key to finding the third zero is: If a polynomial with real coefficients has complex zeros, then they will always come in conjugate pairs. Since we were given one complex zero, 2 - i, then the missing zero must be its conjugate: 2 + i.

\n" ); document.write( "So the three zeros are -2, 2 - i and 2 + i. And when a number, let's call it \"z\", is a zero of a polynomial, then (x - z) is a factor of the polynomial. So, in factored form,
\n" ); document.write( "\"f%28x%29+=+%28x+-+%28-2%29%29%28x+-+%282-i%29%29%28x-%282%2Bi%29%29\"
\n" ); document.write( "Simplifying each factor we get:
\n" ); document.write( "\"f%28x%29+=+%28x+%2B+2%29%28x+-+2%2Bi%29%28x-2-i%29\"

\n" ); document.write( "All that is left is to multiply this out. (Hint for future problems: Multiply the factors with complex conjugate zeros together first.) Multiplying the last two factors can be done with a clever use of the \"%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2\". To show you how I will do some grouping within those factors:
\n" ); document.write( "\"f%28x%29+=+%28x+%2B+2%29%28%28x+-+2%29%2Bi%29%28%28x-2%29-i%29\"
\n" ); document.write( "Treating the \"(x-2)\" as the \"a\" of the pattern and the \"i\" as the \"b\", this pattern tells us that multiplying the last two factors will result in \"a%5E2-b%5E2\":
\n" ); document.write( "\"f%28x%29+=+%28x+%2B+2%29%28%28x+-+2%29%5E2-i%5E2%29\"
\n" ); document.write( "We can use another pattern, \"%28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E2\" to square the (x-2). And \"i%5E2+=+-1\". So we get:
\n" ); document.write( "\"f%28x%29+=+%28x+%2B+2%29%28%28x%29%5E2+-+2%28x%29%282%29%2B%282%5E2%29-%28-1%29%29\"
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "\"f%28x%29+=+%28x+%2B+2%29%28x%5E2+-+4x%2B4%2B1%29\"
\n" ); document.write( "\"f%28x%29+=+%28x+%2B+2%29%28x%5E2+-+4x%2B5%29\"

\n" ); document.write( "Last we multiply the remaining factors (which I will leave to you). Just multiply each term of (x+2) times each term of \"%28x%5E2+-+4x%2B5%29\". (That's 6 multiplications!) Then add any like terms. Then put the terms in standard (highest exponent to lowest exponent) order. \n" ); document.write( "
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