document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176836>Question 1176836</A>: <I><SPAN STYLE=\"word-break: break-all;\"> . Find four rational numbers such that the product of the first, third, and fourth numbers is --6. The second number is 3 less than the first number, the third is 2 less than the second, and the fourth is 2 less than the third </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #804670 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=804670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "f = first number<BR>\n" );
document.write( "s = second number<BR>\n" );
document.write( "t = third number<BR>\n" );
document.write( "h = fourth number<BR>\n" );
document.write( "the letter f was already taken, so I picked the last letter in \"fourth\"<BR>\n" );
document.write( "All of these values are some rational number in the form a/b, where a,b are integers and b is nonzero.\r<BR>\n" );
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document.write( "We are given these four facts<ol><li>The product of the first, third, and fourth numbers is -6.</li><li>The second number is 3 less than the first number</li><li>The third is 2 less than the second</li><li>The fourth is 2 less than the third</li></ol><BR>\n" );
document.write( "Those facts lead to the corresponding equations (in the order shown)<BR>\n" );
document.write( "f*t*h = -6<BR>\n" );
document.write( "s = f-3<BR>\n" );
document.write( "t = s-2<BR>\n" );
document.write( "h = t-2\r<BR>\n" );
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document.write( "I'll refer to them as equations (1) through (4)\r<BR>\n" );
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document.write( "Let's start with equation (4)<BR>\n" );
document.write( "h = t-2<BR>\n" );
document.write( "Now replace t with s-2; this is valid due to equation (3)<BR>\n" );
document.write( "h = t-2<BR>\n" );
document.write( "h = (t)-2<BR>\n" );
document.write( "h = (s-2)-2<BR>\n" );
document.write( "h = s-4<BR>\n" );
document.write( "This tells us the fourth value (h) is 4 less than the second value (variable s).\r<BR>\n" );
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document.write( "We can add 4 to both sides getting<BR>\n" );
document.write( "s = h+4<BR>\n" );
document.write( "Which then can be plugged into equation (2)<BR>\n" );
document.write( "s = f-3<BR>\n" );
document.write( "h+4 = f-3<BR>\n" );
document.write( "h = f-3-4<BR>\n" );
document.write( "h = f-7<BR>\n" );
document.write( "The fourth value is 7 less than the first value\r<BR>\n" );
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document.write( "We have these equations all with h to start off with<BR>\n" );
document.write( "h = f-7<BR>\n" );
document.write( "h = s-4<BR>\n" );
document.write( "h = t-2<BR>\n" );
document.write( "I sorted them so that the one with 'f' goes first, then 's' second, and 't' third\r<BR>\n" );
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document.write( "For each of those three equations shown, isolate the other variable. Doing so leads to these three results<BR>\n" );
document.write( "f = h+7<BR>\n" );
document.write( "s = h+4<BR>\n" );
document.write( "t = h+2\r<BR>\n" );
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document.write( "What this allows us to do is replace the f and t values in equation (1) with something in terms of h. That way we can set up a single variable equation in which we can solve for h\r<BR>\n" );
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document.write( "f*t*h = -6<BR>\n" );
document.write( "(f)*(t)*h = -6<BR>\n" );
document.write( "(h+7)*(h+2)*h = -6<BR>\n" );
document.write( "(h^2+9h+14)*h = -6 ..... FOIL rule<BR>\n" );
document.write( "h^3+9h^2+14h = -6 .... distribute<BR>\n" );
document.write( "h^3+9h^2+14h+6 = 0\r<BR>\n" );
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document.write( "----------------------\r<BR>\n" );
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document.write( "Recall that each f,s,t,h are all rational numbers. <BR>\n" );
document.write( "That must mean the last equation we arrived at has at least one rational root; or else, all the roots of that cubic equation are irrational and that contradicts the instructions. \r<BR>\n" );
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document.write( "The rational root theorem says that we divide the factors of the last term (6) over the factors of the leading coefficient (1)<BR>\n" );
document.write( "List out the plus and minus versions of each ratio\r<BR>\n" );
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document.write( "Luckily that list isn't too big<BR>\n" );
document.write( "If the numerator is 6, then we have the ratios: -6/1, 6/1<BR>\n" );
document.write( "That leads to -6 and 6 respectively.\r<BR>\n" );
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document.write( "If the numerator is 3, then we get -3/1 = -3 and 3/1 = 3<BR>\n" );
document.write( "If the numerator is 2, then we get -2/1 = -2 and 2/1 = 2<BR>\n" );
document.write( "If the numerator is 1, then we get -1/1 = -1 and 1/1 = 1\r<BR>\n" );
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document.write( "The possible rational roots are: <BR>\n" );
document.write( "-6,6<BR>\n" );
document.write( "-3,3<BR>\n" );
document.write( "-2,2<BR>\n" );
document.write( "-1,1\r<BR>\n" );
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document.write( "As you can see, all we've done really is listed all the factors of 6 (positive and negative versions). This is due to the leading coefficient being 1.\r<BR>\n" );
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document.write( "From here, we plug each of those 8 values into the cubic equation<BR>\n" );
document.write( "h^3+9h^2+14h+6 = 0<BR>\n" );
document.write( "to see if the left side becomes 0 or not\r<BR>\n" );
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document.write( "Let's try out h = -6<BR>\n" );
document.write( "h^3+9h^2+14h+6 = (-6)^3+9*(-6)^2+14*(-6)+6 = 30<BR>\n" );
document.write( "That doesn't work. \r<BR>\n" );
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document.write( "Now try h = 6<BR>\n" );
document.write( "h^3+9h^2+14h+6 = (6)^3+9*(6)^2+14*(6)+6 = 630<BR>\n" );
document.write( "That doesn't work either.\r<BR>\n" );
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document.write( "Through trial and error, you should find that only h = -1 works since,<BR>\n" );
document.write( "h^3+9h^2+14h+6 = (-1)^3+9*(-1)^2+14*(-1)+6 = 0\r<BR>\n" );
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document.write( "So h = -1 is a rational root solution to h^3+9h^2+14h+6 = 0\r<BR>\n" );
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document.write( "With that in mind, we can find the other values<BR>\n" );
document.write( "f = h+7 = -1+7 = 6<BR>\n" );
document.write( "s = h+4 = -1+4 = 3<BR>\n" );
document.write( "t = h+2 = -1+2 = 1\r<BR>\n" );
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document.write( "To summarize, we have these values:<BR>\n" );
document.write( "f = 6<BR>\n" );
document.write( "s = 3<BR>\n" );
document.write( "t = 1<BR>\n" );
document.write( "h = -1<BR>\n" );
document.write( "as the first through fourth values in that order.\r<BR>\n" );
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document.write( "----------------------<BR>\n" );
document.write( "Check:\r<BR>\n" );
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document.write( "We'll plug in the values we found for each equation formed at the top of this page.\r<BR>\n" );
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document.write( "f*t*h = -6<BR>\n" );
document.write( "6*1*(-1) = -6<BR>\n" );
document.write( "-6 = -6<BR>\n" );
document.write( "First equation works out\r<BR>\n" );
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document.write( "s = f-3<BR>\n" );
document.write( "3 = 6-3<BR>\n" );
document.write( "3 = 3<BR>\n" );
document.write( "That works out also\r<BR>\n" );
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document.write( "t = s-2<BR>\n" );
document.write( "1 = 3-2<BR>\n" );
document.write( "1 = 1<BR>\n" );
document.write( "Works too\r<BR>\n" );
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document.write( "h = t-2<BR>\n" );
document.write( "-1 = 1-2<BR>\n" );
document.write( "-1 = -1<BR>\n" );
document.write( "We can see that all four original equations work; therefore, the answers have been confirmed.\r<BR>\n" );
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document.write( "----------------------------------------------------<BR>\n" );
document.write( "----------------------------------------------------\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "first = 6<BR>\n" );
document.write( "second = 3<BR>\n" );
document.write( "third = 1<BR>\n" );
document.write( "fourth = -1<BR>\n" );
document.write( "</font> </SPAN>\n" );
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