document.write( "Question 698717: Hi, I need help with a simplification that I need to do every time in a certain type of problem\r
\n" ); document.write( "\n" ); document.write( "I basically need to go from equations in the form $\"+%281%2Bx%29%2F%28%281-x%29%281-3x%29%29+\"$ to equation in the form $\"+%28-1%2F%281-x%29%29%2B%282%2F%281-3x%29%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "I need a detailed way to go from the first to the second\r
\n" ); document.write( "\n" ); document.write( "Another one would be \n" ); document.write( "

Algebra.Com's Answer #431051 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
To go from $\"+%281%2Bx%29%2F%28%281-x%29%281-3x%29%29+\"$ to $\"+A%2F%281-x%29%2BB%2F%281-3x%29\"$
\n" ); document.write( "you need to find the numbers $\"A\"$ and $\"B\"$ that make
\n" ); document.write( " $\"+A%2F%281-x%29%2BB%2F%281-3x%29=%281%2Bx%29%2F%28%281-x%29%281-3x%29%29\"$ true for all $\"x\"$ values.
\n" ); document.write( "You could work from $\"+A%2F%281-x%29%2BB%2F%281-3x%29\"$ to $\"%28%28A%2BB%29%2B%28-3A-B%29x%29%2F%28%281-x%29%281-3x%29%29\"$ ,
\n" ); document.write( "showing how good you are at working with polynomials,
\n" ); document.write( "and then write the equations that make the coefficients in the numerator the same:
\n" ); document.write( " $\"A%2BB=1\"$ and $\"-3A-B=-3\"$
\n" ); document.write( "That would give you a system of linear equations that you could solve for $\"A\"$ and $\"B\"$ .
\n" ); document.write( "
\n" ); document.write( "I prefer a less risky approach, safer from my tendency to mess up work with polynomials.
\n" ); document.write( "Since $\"+A%2F%281-x%29%2BB%2F%281-3x%29=%281%2Bx%29%2F%28%281-x%29%281-3x%29%29\"$ true for all $\"x\"$ values,
\n" ); document.write( "I can use two cunningly-chosen values of $\"x\"$ to generate two linear equations.
\n" ); document.write( "For $\"x=0\"$ , the equation is
\n" ); document.write( " $\"+A%2F%281%29%2BB%2F%281%29=1%2F%28%281%29%281%29%29\"$ or $\"A%2BB=1\"$
\n" ); document.write( "For $\"x=-1\"$ , I get
\n" ); document.write( "

\n" ); document.write( "I do not even write that, confident that I can do that many calculations in my head without messing up, and just write
\n" ); document.write( " $\"+A%2F2%2BB%2F4=0%2F%28%282%29%284%29%29\"$ --> $\"A%2F2%2BB%2F4=0\"$ --> $\"2A%2BB=0\"$
\n" ); document.write( "
\n" ); document.write( "Next I solve $\"system%28A%2BB=1%2C2A%2BB=0%29\"$ to get $\"highlight%28A=-1%29\"$ and $\"highlight%28B=2%29\"$
\n" ); document.write( "
\n" ); document.write( "For $\"A%2F%281-8x%29%2BB%2F%281-10x%29=%281-9x%29%2F%28%281-8x%29%281-10x%29%29\"$ $\"x=0\"$ and $\"x=-1\"$ work well too.
\n" ); document.write( "If they did not, I would use some numbers that made the calculations easier, and did not make the denominators zero.
\n" ); document.write( "For $\"x=0\"$ , the equation is
\n" ); document.write( " $\"A%2F%281%29%2BB%2F%281%29=+1%2F%28%281%29%281%29%29\"$ --> $\"A%2BB=1\"$
\n" ); document.write( "For $\"x=-1\"$ , the equation is
\n" ); document.write( " $\"A%2F%281%2B8%29%2BB%2F%281%2B10%29=+%281%2B9%29%2F%28%281%2B8%29%281%2B10%29%29\"$ --> $\"A%2F9%2BB%2F11=10%2F%289%2A11%29\"$ --> $\"11A%2B9B=10\"$
\n" ); document.write( "Then I solve $\"system%28A%2BB=1%2C11A%2B9B=10%29\"$ to get $\"highlight%28A=1%2F2%29\"$ and $\"highlight%28B=1%2F2%29\"$ \n" ); document.write( "

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