document.write( "Question 278156: (8x^3-4x^2-6x-36)/(x-4) Thanks for your help. \n" ); document.write( "

Algebra.Com's Answer #202451 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%288x%5E3-4x%5E2-6x-36%29%2F%28x-4%29\"$
\n" ); document.write( "Since this fraction does not simplify. I'm guessing that the actual fraction is:
\n" ); document.write( " $\"%288x%5E3-4x%5E2-6x-36%29%2F%28x%5E2-4%29\"$

\n" ); document.write( "Simplifying fractions involves canceling factors that are common to the nnumerator and denominator. To cancel factors we need factors.

\n" ); document.write( "So we start by factoring. The numerator has a GCF of 2 which we can factor out. And the denominator is a difference of squares so it factors easily.
\n" ); document.write( " $\"%282%284x%5E3-2x%5E2-3x-18%29%29%2F%28%28x%2B2%29%28x-2%29%29\"$
\n" ); document.write( "We don't have common factors yet so we keep factoring.
\n" ); document.write( " $\"%282%284x%5E3-2x%5E2-3x-18%29%29%2F%28%28x%2B2%29%28x-2%29%29\"$
\n" ); document.write( "The second factor in the numerator

doesn't fit any of the factoring patterns
has too many terms for trinomial factoring
doesn't appear to be factorable by grouping.

\n" ); document.write( "So it seem that we need to factor by trial and error of the possible rational roots. The possible rational roots are the ratios, positive and negative, which can be formed using a factor of the constant term (at the end, 18 in your case) over a factor of the leading coefficient, 4. The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 4 are 1, 2 and 4. So there are a lot of possible rational roots. But we're only interested in factors that match a factor in the denominator. So the only roots worth trying here are 2 and -2. I'm going to try 2 first. To check a rational root is probably easiest with synthetic division:
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document.write( "2 |  4  -2  -3  -18\r\n" );
document.write( "         8  12   18\r\n" );
document.write( "    ---------------\r\n" );
document.write( "     4  -6   9    0\r\n" );
document.write( "

\n" ); document.write( "The remainder is zero so 2 is a rational root and (x-2) is a factor. And, from the numbers in front of the remainder, the other factor is $\"4x%5E2+-6x+%2B+9\"$ . So now our factored fraction is:
\n" ); document.write( " $\"%282%28x-2%29%284x%5E-6x%2B9%29%29%2F%28%28x%2B2%29%28x-2%29%29\"$
\n" ); document.write( "We could try to continue to factor the numerator but we can see that the rational roots of $\"4x%5E2+-6x+%2B+9\"$ do not include -2. So x+2 will not be a factor. Since any other factors do not help we won't bother factoring any more. Now we can cancel the common factor of x-2:
\n" ); document.write( " $\"%282%284x%5E2-6x%2B9%29%29%2F%28x%2B2%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"%288x%5E2-12x%2B18%29%2F%28x%2B2%29\"$ \n" ); document.write( "

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