document.write( "Question 201523: how do i solve this problem? -12m^3+38m^2-46m-20 \ -3m-1 \n" ); document.write( "
Algebra.Com's Answer #151895 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Since it is just an expression, not an equation, we cannot solve it. But we can simplify it. And how do we simplify a fraction? We either divide it out or we reduce it, if possible, by canceling common factors.
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\n" ); document.write( "\n" ); document.write( "If you know how to do long division with polynomials you will find that it divides out evenly (no remainder)! (Long division of polynomials is too involved for me to explain here.)
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\n" ); document.write( "\n" ); document.write( "If we don't use long division we need to factor the numerator and denominator of \"%28-12m%5E3+%2B+38m%5E2+-46m+-20%29%2F%28-3m-1%29\"

\n" ); document.write( "We start, as always when factoring, with the Greatest Common Factor (GCF). Factoring out -2 from the numerator and -1 from the numerator:
\n" ); document.write( "\"%28-2%2A%286m%5E3+-+19m%5E2+%2B23m+%2B10%29%29%2F%28-1%2A%283m%2B1%29%29\"
\n" ); document.write( "At this point the denominator cannot be factored any further. Factoring \"6m%5E3+-+19m%5E2+%2B23m+%2B10\" further is a bit of a challenge because
  • It is not a trinomial.
  • It does easily factor by grouping.
  • It does not fit any of the common factoring pattens:
    • \"a%5E2+-+b%5E2+=+%28a+%2B+b%29%28a+-+b%29\"
    • \"a%5E3+-+b%5E3+=+%28a+-+b%29%28a%5E2+%2Bab+-+b%5E2%29\"
    • \"a%5E3+%2B+b%5E3+=+%28a+%2B+b%29%28a%5E2+-ab+%2Bb%5E2%29\"
We are left with \"brute force\". The only possible factors of \"6m%5E3+-+19m%5E2+%2B23m+%2B10\" of the form (ax - b) will be those where a is a factor of the leading coefficient, 6, and b is a factor of the constant term, 10. The factors of 6 are 1, 2, 3 and 6 and the factors of 10 are 1, 2, 5 and 10. Not only are there a lot of different combinations of the factors of 6 and factors of 10 we also have to check the negatives of all these combinations!?

\n" ); document.write( "But in your problem we are only interested in factors that will cancel with the denominator. If we are clever we will realize that the only combination worth trying is -1 and 3 (or 1 and -3). To see if this really factors we can use long or synthetic division. I'll attempt to explain synthetic division:
  1. Write down the coefficients of the polynomial horizontally, in order, including the constant term. For your numerator we will write 6 -19 23 10.
  2. Leave a blank line underneath (We'll use it later.)
  3. Draw a line underneath the blank line. (We're going to do some adding.)
  4. Underneath the drawn line copy the first coefficient below itself.
  5. Repeat the following until you reach the end of the numbers:
    1. Multiply the last number by (b/a) and write the result above the line and to the right (underneath the next coefficient)
    2. Add the next coefficient and the product and write the sum below the line
  6. The last sum is the remainder. If it is zero then (ax - b) is a factor and the numbers in front of the 0 are the coefficients of the other factor.


\n" ); document.write( "Keep in mind that we are trying (3m + 1). This means b = -1 and a = 3 and so each multiplication will be by (-1/3). (Note: To show you how this works I am rewriting the whole array at each step. When you do this yourself you just use a single set of 3-4 lines.)
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document.write( "           6  -19  23  10\r\n" );
document.write( "\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6\r\n" );
document.write( "Multiply 6 by (-1/3)\r\n" );
document.write( "           6  -19  23  10\r\n" );
document.write( "               -2\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6\r\n" );
document.write( "Add\r\n" );
document.write( "           6  -19  23  10\r\n" );
document.write( "               -2\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6  -21\r\n" );
document.write( "Multiply -21 by (-1/3)\r\n" );
document.write( "           6  -19  23  10\r\n" );
document.write( "               -2   7\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6  -21\r\n" );
document.write( "Add\r\n" );
document.write( "           6  -19  23  10\r\n" );
document.write( "               -2   7\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6  -21  30\r\n" );
document.write( "Multiply 30 by (-1/3)\r\n" );
document.write( "           6  -19  23  10\r\n" );
document.write( "               -2   7 -10\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6  -21  30\r\n" );
document.write( "Add\r\n" );
document.write( "           6  -19  23  10\r\n" );
document.write( "               -2   7 -10\r\n" );
document.write( "          ---------------\r\n" );
document.write( "           6  -21  30   0\r\n" );
document.write( "

\n" ); document.write( "We have reached the end and the last sum was 0! This means that (3m - (-1)) or (3m + 1) is a factor. It also means that the numbers preceding the 0, 6 -21 30, divided by a are the coefficients of the other factor. The last number is the constant, the one before that is the coefficient of m, the one before that is the coefficient of m^2, etc. So the other factor is (6/3)m^2-(21/3)m+(30/3)

\n" ); document.write( "Your fraction is now:
\n" ); document.write( "\"%28-2%282m%5E2+-+7m+%2B+10%29%283m+%2B+1%29%29%2F%28-1%283m%2B1%29%29\"
\n" ); document.write( "Canceling common factors (3m+1) and (-1) we get
\n" ); document.write( "\"2%282m%5E2+-+7m+%2B+10%29+=+4m%5E2+-14m+%2B+20\"
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