document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=140955>Question 140955</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780321499134/algebrahomeworkh>college algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Can I please get assistance with the following problem?\r<BR>\n" );
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document.write( "factor f(x) into linear factors given that k is a zero of f(x)<BR>\n" );
document.write( "f(x)=x^3-6x^2-25x+150;k=5\r<BR>\n" );
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document.write( "THANK YOU!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #102611 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=102611\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since k=5 is a zero, this means that 5 is a test zero\r<BR>\n" );
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document.write( "So break down <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3-6x%5E2-25x%2B150 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3-6x%5E2-25x%2B150\"   DISABLED_style= \"max-width:90%; height:auto;\" > by using synthetic division\r<BR>\n" );
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document.write( "Set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the function to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 5 by 1 and place the product (which is 5)  right underneath the second  coefficient (which is -6)\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >5</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 5 and -6 to get -1. Place the sum right underneath 5.\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >5</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 5 by -1 and place the product (which is -5)  right underneath the third  coefficient (which is -25)\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >5</TD><TD >-5</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -5 and -25 to get -30. Place the sum right underneath -5.\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >5</TD><TD >-5</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-1</TD><TD >-30</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 5 by -30 and place the product (which is -150)  right underneath the fourth  coefficient (which is 150)\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >5</TD><TD >-5</TD><TD >-150</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-1</TD><TD >-30</TD><TD ></TD></TR>\n" );
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document.write( "    Add -150 and 150 to get 0. Place the sum right underneath -150.\r<BR>\n" );
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document.write( "<TR><TD >5</TD><TD >|</TD><TD >1</TD><TD >-6</TD><TD >-25</TD><TD >150</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >5</TD><TD >-5</TD><TD >-150</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-1</TD><TD >-30</TD><TD >0</TD></TR>\n" );
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document.write( "Since the last column adds to zero, we have a remainder of zero. This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-5 ALIGN=MIDDLE  ALT=\"x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+6x%5E2+-+25x+%2B+150 ALIGN=MIDDLE  ALT=\"x%5E3+-+6x%5E2+-+25x+%2B+150\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the bottom row of coefficients:\r<BR>\n" );
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document.write( "The first 3 coefficients (1,-1,-30) form the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+x+-+30 ALIGN=MIDDLE  ALT=\"x%5E2+-+x+-+30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+-+6x%5E2+-+25x+%2B+150%29%2F%28x-5%29=x%5E2+-+x+-+30 ALIGN=MIDDLE  ALT=\"%28x%5E3+-+6x%5E2+-+25x+%2B+150%29%2F%28x-5%29=x%5E2+-+x+-+30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can use this <a  rel=nofollow HREF=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work\r<BR>\n" );
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document.write( "Basically  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+-+6x%5E2+-+25x+%2B+150 ALIGN=MIDDLE  ALT=\"x%5E3+-+6x%5E2+-+25x+%2B+150\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%28x%5E2+-+x+-+30%29 ALIGN=MIDDLE  ALT=\"%28x-5%29%28x%5E2+-+x+-+30%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "--------------------------------------------------------------------\r<BR>\n" );
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document.write( "Now lets break  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+x+-+30 ALIGN=MIDDLE  ALT=\"x%5E2+-+x+-+30\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-30 ALIGN=MIDDLE  ALT=\"x%5E2-x-30\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <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;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-30 ALIGN=MIDDLE  ALT=\"-30\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and -30 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient -30 to get -30. Now what two numbers multiply to -30 and add to the  middle coefficient -1? Let's list all of the factors of -30:\r<BR>\n" );
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document.write( "Factors of -30:\r<BR>\n" );
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document.write( "1,2,3,5,6,10,15,30\r<BR>\n" );
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document.write( "-1,-2,-3,-5,-6,-10,-15,-30 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to -30\r<BR>\n" );
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document.write( "(1)*(-30)\r<BR>\n" );
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document.write( "(2)*(-15)\r<BR>\n" );
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document.write( "(3)*(-10)\r<BR>\n" );
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document.write( "(5)*(-6)\r<BR>\n" );
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document.write( "(-1)*(30)\r<BR>\n" );
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document.write( "(-2)*(15)\r<BR>\n" );
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document.write( "(-3)*(10)\r<BR>\n" );
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document.write( "(-5)*(6)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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document.write( "Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">-30</TD><TD >1+(-30)=-29</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-15</TD><TD >2+(-15)=-13</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-10</TD><TD >3+(-10)=-7</TD></tr><tr><TD  align=\"center\">5</TD><TD  align=\"center\">-6</TD><TD >5+(-6)=-1</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">30</TD><TD >-1+30=29</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">15</TD><TD >-2+15=13</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">10</TD><TD >-3+10=7</TD></tr><tr><TD  align=\"center\">-5</TD><TD  align=\"center\">6</TD><TD >-5+6=1</TD></tr>\n" );
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document.write( "From this list we can see that 5 and -6 add up to -1 and multiply to -30\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-30 ALIGN=MIDDLE  ALT=\"x%5E2-x-30\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B-6x ALIGN=MIDDLE  ALT=\"5x%2B-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B-6x ALIGN=MIDDLE  ALT=\"5x%2B-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bhighlight%285x%2B-6x%29%2B-30 ALIGN=MIDDLE  ALT=\"x%5E2%2Bhighlight%285x%2B-6x%29%2B-30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x-6x-30 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x-6x-30\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B5x%29%2B%28-6x-30%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B5x%29%2B%28-6x-30%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B5%29-6%28x%2B5%29 ALIGN=MIDDLE  ALT=\"x%28x%2B5%29-6%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-6%29%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x-6%29%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B5 ALIGN=MIDDLE  ALT=\"x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x-6x-30 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x-6x-30\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-6%29%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x-6%29%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-30 ALIGN=MIDDLE  ALT=\"x%5E2-x-30\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-6%29%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x-6%29%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-6x%5E2-25x%2B150 ALIGN=MIDDLE  ALT=\"x%5E3-6x%5E2-25x%2B150\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%28x-6%29%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x-5%29%28x-6%29%28x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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