document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=74388>Question 74388</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Any help here I am the worst with math problems.\r<BR>\n" );
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document.write( "Find the product of (x2 - 3x + 5) with the quotient of (18x6 - 27x5 - 9x3) ÷ 9x3. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #53413 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=53413\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the product of (x^2 - 3x + 5) with the quotient of (18x^6 - 27x^5 - 9x^3) ÷ 9x^3. <BR>\n" );
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document.write( "First we'll work on finding the quotient of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2818x%5E6+-+27x%5E5+-+9x%5E3%29%2F+9x%5E3 ALIGN=MIDDLE  ALT=\"%2818x%5E6+-+27x%5E5+-+9x%5E3%29%2F+9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You do that by dividing the denominator into each of the terms in the numerator, so your<BR>\n" );
document.write( "answer has three terms.  And the way you do the division is you divide the 9 into the number<BR>\n" );
document.write( "in the term and you also subtract the exponents on their exponents of x.  So to divide<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3 ALIGN=MIDDLE  ALT=\"9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%5E6 ALIGN=MIDDLE  ALT=\"18x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > you divide the 9 into the 18 to get 2, and then you subtract<BR>\n" );
document.write( "the exponents of the x terms to get as the x part of the answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%286-3%29+=+x%5E3 ALIGN=MIDDLE  ALT=\"x%5E%286-3%29+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Putting this together results in the division of the first term by the denominator <BR>\n" );
document.write( "being <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3 ALIGN=MIDDLE  ALT=\"2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We use the same process to divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-27x%5E5 ALIGN=MIDDLE  ALT=\"-27x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3 ALIGN=MIDDLE  ALT=\"9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Divide the 9 into the -27 and<BR>\n" );
document.write( "get -3.  Then divide the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5 ALIGN=MIDDLE  ALT=\"x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%285-3%29+=+x%5E2 ALIGN=MIDDLE  ALT=\"x%5E%285-3%29+=+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Put these two divisions together and the answer for the second term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2 ALIGN=MIDDLE  ALT=\"-3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "And finally, divide the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E3 ALIGN=MIDDLE  ALT=\"9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the third term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9x%5E3 ALIGN=MIDDLE  ALT=\"-9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The 9 into the -9 results<BR>\n" );
document.write( "in -1 and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > results in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%283-3%29+=+x%5E0 ALIGN=MIDDLE  ALT=\"x%5E%283-3%29+=+x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and by definition<BR>\n" );
document.write( "anything raised to the 0 power is 1.  So the answer for this division is -1 times 1 or<BR>\n" );
document.write( "just -1.<BR>\n" );
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document.write( "Putting all three terms together, the quotient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-3x%5E2+-1 ALIGN=MIDDLE  ALT=\"2x%5E3+-3x%5E2+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Halfway home to being<BR>\n" );
document.write( "done.<BR>\n" );
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document.write( "Next the problem requires that this quotient be multiplied by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can do<BR>\n" );
document.write( "this by picking one term in the quotient and multiplying it by all the terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You will get 3 answers.  Then you pick a second term from the quotient and multiply it<BR>\n" );
document.write( "by all three terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 3 more answers.  Finally, you take the final<BR>\n" );
document.write( "term in the quotient and multiply it by all three terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the final <BR>\n" );
document.write( "group of 3 answers.  Then you add all 9 answers together, look for common terms that can be <BR>\n" );
document.write( "combined, combine them and what you end up with is the answer to the problem.<BR>\n" );
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document.write( "Before we begin this process, let's establish the multiplication ground rules.  Just the opposite<BR>\n" );
document.write( "of the division process.  This time we multiply the numbers, not divide them. And we add<BR>\n" );
document.write( "the exponents, not subtract them.<BR>\n" );
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document.write( "OK, let's go.  From the quotient let's first take the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3 ALIGN=MIDDLE  ALT=\"2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we'll multiply it<BR>\n" );
document.write( "by all the terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The first multiplication is of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3 ALIGN=MIDDLE  ALT=\"2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A1 ALIGN=MIDDLE  ALT=\"2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%283%2B2%29+=+x%5E5 ALIGN=MIDDLE  ALT=\"x%5E%283%2B2%29+=+x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to give us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E5 ALIGN=MIDDLE  ALT=\"2x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The next multiplication is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3 ALIGN=MIDDLE  ALT=\"2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x ALIGN=MIDDLE  ALT=\"-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and its answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28-3%29 ALIGN=MIDDLE  ALT=\"2%2A%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%283%2B1%29+=+x%5E4 ALIGN=MIDDLE  ALT=\"x%5E%283%2B1%29+=+x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which gives us a total result of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6x%5E4 ALIGN=MIDDLE  ALT=\"-6x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And finally we <BR>\n" );
document.write( "multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3 ALIGN=MIDDLE  ALT=\"2x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B5 ALIGN=MIDDLE  ALT=\"%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which results in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A5 ALIGN=MIDDLE  ALT=\"2%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%283%2B0%29+=+x%5E3 ALIGN=MIDDLE  ALT=\"x%5E%283%2B0%29+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for<BR>\n" );
document.write( "an answer of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%5E3 ALIGN=MIDDLE  ALT=\"10x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  We now have the first 3 of the 9 multiplication answers and<BR>\n" );
document.write( "they are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E5+-+6x%5E4+%2B10x%5E3 ALIGN=MIDDLE  ALT=\"2x%5E5+-+6x%5E4+%2B10x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Next we take the second term from the quotient.  It is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2 ALIGN=MIDDLE  ALT=\"-3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we multiply it <BR>\n" );
document.write( "by all three terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Maybe by now you have enough of the idea to see <BR>\n" );
document.write( "that the three answers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E4+%2B+9x%5E3+-15x%5E2 ALIGN=MIDDLE  ALT=\"-3x%5E4+%2B+9x%5E3+-15x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. (Multiply the coefficients and add the <BR>\n" );
document.write( "exponents for each term.)<BR>\n" );
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document.write( "Finally multiply the last term of the quotient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and multiply it by each term in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x+%2B+5 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2+%2B+3x+-+5 ALIGN=MIDDLE  ALT=\"-x%5E2+%2B+3x+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > for these 3 answers.<BR>\n" );
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document.write( "So in one long string the answers are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E5+-+6x%5E4+%2B+10x%5E3+-+3x%5E4+%2B+9x%5E3+-+15x%5E2+-+x%5E2+%2B+3x+-5 ALIGN=MIDDLE  ALT=\"2x%5E5+-+6x%5E4+%2B+10x%5E3+-+3x%5E4+%2B+9x%5E3+-+15x%5E2+-+x%5E2+%2B+3x+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You have 2 terms containing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. They are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6x%5E4 ALIGN=MIDDLE  ALT=\"-6x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E4 ALIGN=MIDDLE  ALT=\"-3x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Combine them<BR>\n" );
document.write( "into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9x%5E4 ALIGN=MIDDLE  ALT=\"-9x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  You also can combine the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B10x%5E3+%2B+9x%5E3+=+19x%5E3 ALIGN=MIDDLE  ALT=\"%2B10x%5E3+%2B+9x%5E3+=+19x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And you <BR>\n" );
document.write( "can combine the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x%5E2+-x%5E2+=+-16x%5E2 ALIGN=MIDDLE  ALT=\"-15x%5E2+-x%5E2+=+-16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  None of the other terms can<BR>\n" );
document.write( "be combined so the resulting string of terms is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E5+-9x%5E4+%2B+19x%5E3+-+16x%5E2+%2B3x+-5 ALIGN=MIDDLE  ALT=\"2x%5E5+-9x%5E4+%2B+19x%5E3+-+16x%5E2+%2B3x+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that should be the answer to this problem.<BR>\n" );
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document.write( "It's been a long journey.  I hope that this exercise has given you some insight into multiplying<BR>\n" );
document.write( "and dividing algebraic terms and how you combine terms that have like powers of x. </SPAN>\n" );
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