document.write( "Question 801325: Find a 3rd degree polynomial with integral coefficients that has 2 +i and -3 as two of its roots. \n" ); document.write( "

Algebra.Com's Answer #483516 by josgarithmetic(39623) $\"\"$ $\"About$

You can put this solution on YOUR website!
Roots given ALSO include 2-i, the conjugate of 2+i.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Form the binomial factors of the polynomial based on the now known roots:
\n" ); document.write( "x+3, (x-(2+i)), (x-(2-i)).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Wanting only general form polynomial with integer coefficients, work with first the two complex factors with the imaginary components:
\n" ); document.write( " $\"%28x-%282%2Bi%29%29%28x-%282-i%29%29\"$
\n" ); document.write( " $\"%28x-2-i%29%28x-2%2Bi%29\"$
\n" ); document.write( " $\"%28%28x-2%29-i%29%28%28x-2%29%2Bi%29\"$
\n" ); document.write( "Recognize the factors giving the difference of two squares.
\n" ); document.write( " $\"%28x-2%29%5E2-i%5E2\"$
\n" ); document.write( " $\"x%5E2-4x%2B4%2B1\"$
\n" ); document.write( " $\"x%5E2-4x%2B5\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Still in factored form but showing only integer coefficients within each polynomial factor, polynomial is $\"highlight%28%28x%2B3%29%28x%5E2-4x%2B5%29%29\"$ which you can further multiply if you need. \n" ); document.write( "

\n" );