document.write( "Question 1209677: When g(x) is divided by x^2 - x - 6, the remainder is 2x + 7. What is the value
\n" ); document.write( "of g(8)? \n" ); document.write( "
Algebra.Com's Answer #849800 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "There isn't enough info given.\r
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\n" ); document.write( "\n" ); document.write( "Consider
\n" ); document.write( "g(x) = x^3+6x^2-11x-35
\n" ); document.write( "h(x) = x^2-x-6\r
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\n" ); document.write( "\n" ); document.write( "Use polynomial long division to determine that,
\n" ); document.write( "g/h = (x+7) remainder (2x+7)
\n" ); document.write( "We fulfill the requirement of dividing a function g(x) over h(x) = x^2-x-6 where it yields remainder 2x+7.
\n" ); document.write( "There are many online calculators that will perform polynomial long division (some even provide the step-by-step guide).
\n" ); document.write( "I used the division tool in GeoGebra
\n" ); document.write( "Then,
\n" ); document.write( "g(x) = x^3+6x^2-11x-35
\n" ); document.write( "g(8) = (8)^3+6*(8)^2-11*(8)-35
\n" ); document.write( "g(8) = 773\r
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\n" ); document.write( "\n" ); document.write( "Now consider,
\n" ); document.write( "g(x) = x^4-6x^3+12x^2+19x-71
\n" ); document.write( "h(x) = x^2-x-6
\n" ); document.write( "The first function has been changed, but the h(x) function is the same as before.
\n" ); document.write( "You should find that,
\n" ); document.write( "g/h = (x^2-5x+13) remainder (2x+7)
\n" ); document.write( "We get the same remainder as before.
\n" ); document.write( "However,
\n" ); document.write( "g(x) = x^4-6x^3+12x^2+19x-71
\n" ); document.write( "g(8) = (8)^4-6*(8)^3+12*(8)^2+19*(8)-71
\n" ); document.write( "g(8) = 1873
\n" ); document.write( "This contradicts the previous result g(8) = 773
\n" ); document.write( "This means we simply do not have enough information to determine g(8)\r
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\n" ); document.write( "\n" ); document.write( "I have a feeling that your teacher made a typo when asking about g(8)\r
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\n" ); document.write( "\n" ); document.write( "Despite not being able to find g(8) we can find either g(-2) or g(3)
\n" ); document.write( "Note that -2 and 3 are roots of x^2-x-6 = (x+2)(x-3) = 0
\n" ); document.write( "The fact we get 0 is important.\r
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\n" ); document.write( "\n" ); document.write( "Here's the scratch work to finding g(3)
\n" ); document.write( "g(x) = unknown
\n" ); document.write( "h(x) = x^2-x-6
\n" ); document.write( "h(3) = 3^2-3-6 = 0
\n" ); document.write( "g(x)/h(x) = quotient + remainder/h(x)
\n" ); document.write( "g(x) = h(x)*quotient + remainder
\n" ); document.write( "g(x) = h(x)*quotient + 2x+7
\n" ); document.write( "g(3) = h(3)*quotient + 2(3)+7
\n" ); document.write( "g(3) = 0*quotient + 2(3)+7
\n" ); document.write( "g(3) = 0 + 2(3)+7
\n" ); document.write( "g(3) = 13\r
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\n" ); document.write( "\n" ); document.write( "Pay close attention to the fact that the h(3)*quotient portion goes to 0 since h(3) = 0.
\n" ); document.write( "This allows us to ignore the quotient entirely.
\n" ); document.write( "This only happens for x = -2 or x = 3. Unfortunately it does not happen when x = 8.\r
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\n" ); document.write( "\n" ); document.write( "Through similar steps, we can find g(-2) = 3
\n" ); document.write( "This assumes that the x^2-x-6 and 2x+7 portions aren't typos.
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