document.write( "Question 1123541: Let f(m) = m˄5 - 11m˄3 - 26m˄2 + 48m + 144. Given that m = -2 + 2i and m = -2 are
\n" ); document.write( "roots of f(m), find all the other roots of f(m) and write f(m) as a product of irreducible real quadratic and linear functions. \n" ); document.write( "
Algebra.Com's Answer #739883 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "I would go about this in a different order....

\n" ); document.write( "Given the root of -2, first use synthetic division to remove that root.
\r\n" );
document.write( "\r\n" );
document.write( "  -2 |  1  0  -11  -26  48  144\r\n" );
document.write( "     |    -2    4   14  24 -144\r\n" );
document.write( "     --------------------------\r\n" );
document.write( "        1 -2   -7  -12  72   0\r\n" );
document.write( "

\n" ); document.write( "At this point we know

\n" ); document.write( "\"m%5E5+-+11m%5E3-26m%5E2%2B48m%2B144+=+%28m%2B2%29%28m%5E4-2m%5E3-7m%5E2-12m%2B72%29\"

\n" ); document.write( "Next, given the root -2+2i, we know -2-2i is also a root, because complex roots occur in conjugate pairs.

\n" ); document.write( "We can get the quadratic factor corresponding to that pair of roots by using the fact that in the quadratic equation x^2+bx+c=0 the sum of the roots is -b and the product is c.

\n" ); document.write( "The sum of these two roots is -4; their product is 4-4i^2 = 4+4 = 8. So the quadratic factor corresponding to these two roots is m^2+4m+8.

\n" ); document.write( "So now we know

\n" ); document.write( "\"m%5E5+-+11m%5E3-26m%5E2%2B48m%2B144+=+%28m%2B2%29%28m%5E2%2B4m%2B8%29%28m%5E2%2Bam%2Bb%29\"

\n" ); document.write( "where the coefficients a and b in the second quadratic factor are yet to be determined.

\n" ); document.write( "To find those coefficients, we know that

\n" ); document.write( "\"%28m%5E4-2m%5E3-7m%5E2-12m%2B72%29=%28m%5E2%2B4m%2B8%29%28m%5E2%2Bam%2Bb%29\"}

\n" ); document.write( "We can immediately see that b=9 by looking at the constant term: 72 is equal to 8 times b.

\n" ); document.write( "And one quick way (with a little practice) to find the coefficient a is to see that the coefficient of the m^3 term, -2, comes from the two partial products (m^2)*(am) and (4m)(m^2). So

\n" ); document.write( "\"-2+=+a%2B4\"
\n" ); document.write( "\"a+=+-6\"

\n" ); document.write( "So now we know the factorization is

\n" ); document.write( "\"m%5E5+-+11m%5E3-26m%5E2%2B48m%2B144+=+%28m%2B2%29%28m%5E2%2B4m%2B8%29%28m%5E2-6m%2B9%29\"

\n" ); document.write( "And finally we see that the second quadratic factor is reducible, and the final complete factorization is

\n" ); document.write( "\"m%5E5+-+11m%5E3-26m%5E2%2B48m%2B144+=+%28m%2B2%29%28m%5E2%2B4m%2B8%29%28m-3%29%5E2\" \n" ); document.write( "
\n" );