document.write( "Question 1126012: Find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and zeros of 3 and 2 plus i .
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Algebra.Com's Answer #742372 by greenestamps(13215) $\"\"$ $\"About$

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\n" ); document.write( "Given the three roots p, q, and r, one way you can find the equation is multiply out (x-p)(x-q)(x-r), as shown by the other tutor. With the two complex roots, the algebra gets a bit ugly.

\n" ); document.write( "So here is another way to find the equation that some students might find easier.

\n" ); document.write( "In the final equation (with leading coefficient 1)

\n" ); document.write( " $\"f%28x%29+=+x%5E3%2Bbx%5E2%2Bcx%2Bd\"$ ,

\n" ); document.write( "the coefficient of the quadratic term is the opposite of the sum of the roots:
\n" ); document.write( " $\"b+=+-%28p%2Bq%2Br%29\"$

\n" ); document.write( "the coefficient of the linear term is the sum of the products of the roots two at a time:
\n" ); document.write( " $\"c+=+pq%2Bpr%2Bqr\"$

\n" ); document.write( "and the constant term is the opposite of the product of the three roots:
\n" ); document.write( " $\"d+=+-pqr\"$

\n" ); document.write( "With roots 3, 2+i, and 2-i, we have...

\n" ); document.write( " $\"p%2Bq%2Br+=+3%2B%282%2Bi%29%2B%282-i%29+=+7\"$ so b = -7;

\n" ); document.write( " $\"pq%2Bpr%2Bqr+=+%286%2B3i%29%2B%286-3i%29%2B5+=+17\"$ so c = 17; and

\n" ); document.write( " $\"pqr+=+3%285%29+=+15\"$ so d = -15.

\n" ); document.write( "Then the polynomial is

\n" ); document.write( " $\"f%28x%29+=+x%5E3-7x%5E2%2B17x-15\"$ \n" ); document.write( "

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