document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189262>Question 1189262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a) Use the remainder theorem to find the remainder when 𝑓(𝑥)=−4𝑥^3 +5𝑥^2 + 8 is <BR>\n" );
document.write( "   divided by 𝑥+3. Then use the Factor Theorem to determine whether 𝑥  + 3 is a <BR>\n" );
document.write( "   factor of 𝑓(𝑥).\r<BR>\n" );
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document.write( "b) Use the Rational Zeros Theorem to find all the real zeros of the polynomial <BR>\n" );
document.write( "   function 𝑓(𝑥)=x^4 -x^3 -6x^2 +4x +8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820577 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=820577\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "Remainder theorem: Dividing p(x) over (x-k) yields some quotient q(x) and a remainder p(k).<BR>\n" );
document.write( "Special case: if the remainder is 0, then (x-k) is a factor of p(x).\r<BR>\n" );
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document.write( "Compare (x+3) with (x-k) to see that k = -3<BR>\n" );
document.write( "It might help to rewrite x+3 as x-(-3).\r<BR>\n" );
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document.write( "f(x) = -4x^3+5x^2+8<BR>\n" );
document.write( "f(-3) = -4(-3)^3+5(-3)^2+8<BR>\n" );
document.write( "f(-3) = 161\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "The remainder is 161<BR>\n" );
document.write( "(x+3) is <u>not</u> a factor of f(x) because we didn't get a remainder of 0.\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Due to the leading coefficient being 1, this means we simply need to list the plus/minus version of each factor of the last term 8\r<BR>\n" );
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document.write( "The list of all possible rational roots:<BR>\n" );
document.write( "1, -1, 2, -2, 4, -4, 8, -8\r<BR>\n" );
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document.write( "Then check each value to see if we get f(x) = 0 or not. \r<BR>\n" );
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document.write( "If we tried x = 1, then,<BR>\n" );
document.write( "f(x) = x^4-x^3-6x^2+4x+8<BR>\n" );
document.write( "f(1) = (1)^4-(1)^3-6(1)^2+4(1)+8<BR>\n" );
document.write( "f(1) = 6<BR>\n" );
document.write( "We don't get a result of 0, which means x = 1 is <u>not</u> a root or zero of the function.\r<BR>\n" );
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document.write( "Trying x = -1 leads to:<BR>\n" );
document.write( "f(x) = x^4-x^3-6x^2+4x+8<BR>\n" );
document.write( "f(-1) = (-1)^4-(-1)^3-6(-1)^2+4(-1)+8<BR>\n" );
document.write( "f(-1) = 0<BR>\n" );
document.write( "We get zero this time, so x = -1 is a root of f(x)\r<BR>\n" );
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document.write( "Repeat these steps for the remaining possible roots listed above. You should find that only the following are roots: -2, -1, 2<BR>\n" );
document.write( "Side note: x = 2 is a double root\r<BR>\n" );
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document.write( "Answer: -2, -1, and 2 are zeros of f(x)<BR>\n" );
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