document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92556>Question 92556</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0534494498/algebrahomeworkh>Algebra and Trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Find the polynomial f(x) of degree three that has zeroes at 1,2, and 4 such that f(0)= -16.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67392 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=67392\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Being a zero of a polynomial means when you assign a value to x in the polynomial, the polynomial<BR>\n" );
document.write( "will have a value of zero.<BR>\n" );
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document.write( "Suppose we could form a polynomial that has three values of x for which the polynomial will<BR>\n" );
document.write( "have a value of zero. A way that we could do that is to make the polynomial be the product of<BR>\n" );
document.write( "three factors ... the factors having the form (x + a), (x + b), and (x + c).  The polynomial<BR>\n" );
document.write( "would be the product of these three factors. In other words, the polynomial would be:<BR>\n" );
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document.write( "(x + a)*(x + b)*(x + c)<BR>\n" );
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document.write( "But think about this ... the value of the polynomial would be zero if any one of the factors<BR>\n" );
document.write( "were zero because zero times the other two factors makes the entire polynomial zero. <BR>\n" );
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document.write( "We were given three values of x that will make the polynomial go to zero.  The first value<BR>\n" );
document.write( "was x = 1.  Let's use the first factor (x + a) for this. If x equals 1 in this factor<BR>\n" );
document.write( "and we want that factor to go to be able to go to zero, then a must equal -1.  So our first<BR>\n" );
document.write( "factor is (x - 1). Similarly, we want the second factor to go to zero if x = 2. For it to<BR>\n" );
document.write( "go to zero when x = 2, then b must equal -2. So our second factor is (x - 2). Similarly,<BR>\n" );
document.write( "the third factor needs to go to zero when x is equal to 4. For this to happen, c must be<BR>\n" );
document.write( "equal to -4, so the third factor is (x - 4).<BR>\n" );
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document.write( "We can now get these three factors in polynomial form by multiplying them together:<BR>\n" );
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document.write( "(x - 1)*(x - 2)*(x - 4)<BR>\n" );
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document.write( "You can do that multiplication in two stages. First multiply (x - 1) times (x - 2) and then <BR>\n" );
document.write( "multiply the resulting product by (x - 4) to get the answer.  Assuming you can do the multiplication,<BR>\n" );
document.write( "here's the two results for you to use as checks:<BR>\n" );
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document.write( "(x - 1)*(x - 2) = x^2 - 3x + 2<BR>\n" );
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document.write( "Then doing the second stage multiplication:<BR>\n" );
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document.write( "(x^2 - 3x + 2)*(x - 4) = x^3 - 7x^2 + 14x - 8<BR>\n" );
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document.write( "You can check this by letting x = 1 first and this polynomial becomes:<BR>\n" );
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document.write( "1^3 - 7(1^2) + 14(1) - 8 = 1 - 7 + 14 - 8 = 0<BR>\n" );
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document.write( "Next let x = 2 and the polynomial becomes<BR>\n" );
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document.write( "2^3 - 7(2^2) + 14(2) - 8 = 8 - 28 + 28 - 8 = 0<BR>\n" );
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document.write( "Finally, let x = 4 and the polynomial becomes:<BR>\n" );
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document.write( "4^3 - 7(4^2) + 14(4) - 8 = 64 - 112 + 56 - 8 = 0<BR>\n" );
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document.write( "So our three zeros work out OK. But we still have another criterion to meet. That criterion<BR>\n" );
document.write( "says that when x equals zero, the polynomial must equal -16. Well, our polynomial <BR>\n" );
document.write( "does not meet that. Our polynomial is<BR>\n" );
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document.write( "x^3 - 7x^2 + 14x - 8<BR>\n" );
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document.write( "and when x is equal to zero, all the terms containing x disappear so we are just left with<BR>\n" );
document.write( "a value of -8.<BR>\n" );
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document.write( "But what happens if we double our polynomial ... multiply all of its terms by 2. That would<BR>\n" );
document.write( "mean that our polynomial would be formed from the factors:<BR>\n" );
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document.write( "2*(x - 1)*(x - 2)*(x - 4)<BR>\n" );
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document.write( "It would still go to zero if x = 1 or if x = 2 or if x = 4, so that property remains unchanged.<BR>\n" );
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document.write( "But we already know that (x - 1)*(x - 2)*(x - 4) = x^3 - 7x^2 + 14x - 8<BR>\n" );
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document.write( "And if we multiply this by 2 the resulting polynomial is:<BR>\n" );
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document.write( "2x^3 - 14x^2 + 28x - 16<BR>\n" );
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document.write( "Since this still contains the factors (x - 1), (x - 2), and (x - 4) so it goes to zero<BR>\n" );
document.write( "when x = 1 or when x = 2 or when x = 4.  But what about when x = 0?  In the polynomial<BR>\n" );
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document.write( "2x^3 - 14x^2 + 28x - 16<BR>\n" );
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document.write( "if x is set to zero then all the terms containing x are zero and we are left with the<BR>\n" );
document.write( "value of the polynomial being -16, just as the problem requires it should be.<BR>\n" );
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document.write( "So the answer is that the polynomial is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+2x%5E3+-+14x%5E2+%2B+28x+-+16 ALIGN=MIDDLE  ALT=\"f%28x%29+=+2x%5E3+-+14x%5E2+%2B+28x+-+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We have already checked when x equals zero to find that f(0) = -16. You can further<BR>\n" );
document.write( "check the value of f(1), f(2), and f(4) to make sure all three of them are zero.<BR>\n" );
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document.write( "Hope this helps you to understand the problem and how it can be solved.  </SPAN>\n" );
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