document.write( "Question 1157886: Use the Factor Theorem to find all real zeros for the given polynomial function and one factor. (Enter your answers as a comma-separated list.)
\n" ); document.write( "f(x) = 8x3 + x2 − 55x + 42; x + 3 \n" ); document.write( "
Algebra.Com's Answer #780795 by greenestamps(13208)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Strange wording of the problem -- it asks to FIND one factor; but it already SHOWS one factor....

\n" ); document.write( "(1) Show that x+3 is a factor using synthetic division:
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document.write( "  -3  |  8    1  -55   42\r\n" );
document.write( "      |\r\n" );
document.write( "      +-------------------\r\n" );
document.write( "\r\n" );
document.write( "  -3  |  8    1  -55   42\r\n" );
document.write( "      |     -24\r\n" );
document.write( "      +-------------------\r\n" );
document.write( "         8  -23\r\n" );
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document.write( "  -3  |  8    1  -55   42\r\n" );
document.write( "      |     -24   69\r\n" );
document.write( "      +-------------------\r\n" );
document.write( "         8  -23   14\r\n" );
document.write( "\r\n" );
document.write( "  -3  |  8    1  -55   42\r\n" );
document.write( "      |     -24   69  -42\r\n" );
document.write( "      +-------------------\r\n" );
document.write( "         8  -23   14    0

\n" ); document.write( "(x+3) is a root; and

\n" ); document.write( "\"8x%5E3%2Bx%5E2-55x%2B42+=+%28x%2B3%29%288x%5E2-23x%2B14%29\"

\n" ); document.write( "There are many methods for factoring that quadratic; I am old school and think the best thing to do is try to use logical reasoning to find the pair of linear factors.

\n" ); document.write( "With this one, that leads quickly to the correct factorization.

\n" ); document.write( "We have two choices for the form of the factorization:
\n" ); document.write( "(4x-?)(2x-)
\n" ); document.write( "(8x-?)(x-?)

\n" ); document.write( "But logical reasoning quickly tells me the first is impossible, because the middle term of the product is going to be even.

\n" ); document.write( "So the factorization is of the form (8x-?)(x-?); quick trial with the factors of 14 leads us to (8x-7)(x-2).

\n" ); document.write( "The complete factorization is

\n" ); document.write( "\"8x%5E3%2Bx%5E2-55x%2B42+=+%28x%2B3%29%288x-7%29%28x-2%29\"

\n" ); document.write( "The roots are -3, 7/8, and 2.
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