document.write( "Question 882930: a Polynomial P(x) and a divisor d(x) are given. Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x), and express P(x) in the form d(x)*Q(x)+R(x)\r
\n" ); document.write( "\n" ); document.write( "\"P%28x%29=+x%5E3-x%5E2%2B7\"
\n" ); document.write( "\"d%28x%29=+x%2B2\" \n" ); document.write( "
Algebra.Com's Answer #533358 by KMST(5328)\"\" \"About 
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THE SOLUTION:
\n" ); document.write( "\"highlight%28Q%28x%29=x%5E2-3x%2B6%29\" , \"highlight%28R%28x%29=-5%29\" , and P(x) in the form d(x)*Q(x)+R(x), or
\n" ); document.write( "\"P%28x%29=d%28x%29%2AQ%28x%29%2BR%28x%29\" , is \"highlight%28P%28x%29=%28x%2B2%29%28x%5E2-3x%2B6%29-5%29\" .
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\n" ); document.write( "THE LONG DIVISION:
\n" ); document.write( "The format for long division varies from country to country (and maybe even from teacher to teacher. I will use the format my children were taught in the USA.
\n" ); document.write( "The first term of quotient \"Q%28x%29\" is the quotient of the first terms \"x%5E3%2Fx=red%28x%5E2%29\" .
\n" ); document.write( "Then \"red%28x%5E2%29%2Ad%28x%29=red%28x%5E2%29%28x%2B2%29=x%5E2%2B2x%5E2\" is subtracted from \"P%28x%29=+x%5E3-x%5E2%2B7\" .
\n" ); document.write( "Subtracting \"x%5E2%2B2x%5E2\" is adding \"%28-1%29%28x%5E2%2B2x%5E2%29=-x%5E2-2x%5E2\" ,
\n" ); document.write( "and the result is
\n" ); document.write( "\"P%28x%29%2B%28-x%5E2-2x%5E2%29=x%5E3-x%5E2%2B7-x%5E2-2x%5E2=-3x%5E2%2B7\" .
\n" ); document.write( "That is shown below, but I am saving room for terms in \"x\" by including the term \"%220%22\"\"x\" .
\n" ); document.write( "--->
\n" ); document.write( "We continue dividing by \"x%2B2\" the remaining \"-3x%5E2%2B7\" ,
\n" ); document.write( "which I wrote as \"-3x%5E2%2B0x%2B7\" to save room for the term in \"x\" .
\n" ); document.write( "The next term in the quotient, which comes from dividing first terms, is
\n" ); document.write( "\"%28-3x%5E2%29%2Fx=red%28-3x%29\" .
\n" ); document.write( "That term, times the divisor is
\n" ); document.write( "\"%28red%28-3x%29%29%2Ad%28x%29=%28red%28-3x%29%29%28x%2B2%29=-3x%5E2-6x\" , which must be subtracted from
\n" ); document.write( "\"-3x%5E2%2B7=-3x%5E2%2B0x%2B7\" .
\n" ); document.write( "Subtracting \"-3x%5E2-6x\" is adding \"%28-1%29%28-3x%5E2-6x%29=3x%5E2%2B6x\" ,
\n" ); document.write( "and the result is \"-3x%5E2%2B7%2B3x%5E2%2B6x=6x%2B7\" .
\n" ); document.write( "--->
\n" ); document.write( "To find the next term of \"Q%28x%29\",
\n" ); document.write( "we have to divide the first term of the remaining \"6x%2B7\"
\n" ); document.write( "by the first term of divisor \"d%28x%29=x%2B2\" :
\n" ); document.write( "\"6x%2Fx=red%286%29\" .
\n" ); document.write( "So the last term of the quotient will be \"%22%2B%22\"\"red%286%29\" ,
\n" ); document.write( "and we need to subtract the product
\n" ); document.write( "\"red%286%29%2Ad%28x%29=red%2A%286%29%2A%28x%2B2%29=6x%2B12\" from the remaining \"6x%2B7\" .
\n" ); document.write( "Since subtracting \"6x%2B12\" is adding \"%28-1%29%286x%2B12%29=-6x-12\" ,
\n" ); document.write( "I add \"-6x-12\" to that remaining \"6x%2B7\" and find the remainder:
\n" ); document.write( "\"R%28x%29=6x%2B7%2B%28-6x-12%29=6x%2B7%2B-6x-12=highlight%28red%28-5%29%29\" :
\n" ); document.write( "
\n" ); document.write( "---> \n" ); document.write( "
\n" );