document.write( "Question 1173907: Given these key features:
\n" ); document.write( "f(0) = 5
\n" ); document.write( "leading coefficient is -2
\n" ); document.write( "zeroes at -3, -1, 4
\n" ); document.write( "Create two possible algebraic models, using two different degrees of polynomial functions.
\n" ); document.write( "Express both in standard and factored form. \n" ); document.write( "
Algebra.Com's Answer #799275 by greenestamps(13216)\"\" \"About 
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\n" ); document.write( "I would be curious to know if the requirements as you show them are really what was intended. It makes for strange polynomials.

\n" ); document.write( "And if -3, -1, and 4 are to be the ONLY zeros, then there are no polynomials that satisfy all the conditions.

\n" ); document.write( "The solutions are going to be complicated even in factored form; if you really need them in standard form, I'll let you do that part.

\n" ); document.write( "Let's start with leading coefficient 2 and zeros at -3, -1, and 4. The basic polynomial is

\n" ); document.write( "\"p%28x%29+=+-2%28x%2B3%29%28x%2B1%29%28x-4%29\"

\n" ); document.write( "\"graph%28400%2C400%2C-5%2C5%2C-50%2C50%2C-2%28x%2B3%29%28x%2B1%29%28x-4%29%29\"

\n" ); document.write( "For that polynomial, f(0)=24; but the requirement is f(0)=5.

\n" ); document.write( "To get a polynomial with the given zeros and leading coefficient, and also with f(0)=5, we need to add further linear factors of the form (x+a), (x+b), ..., where the product of ab... is 5/24.

\n" ); document.write( "That will make f(0)=(5/24)(24)=5, as required; but of course it will add more zeros to the function.

\n" ); document.write( "We can do it with one more factor of (x+5/24):

\n" ); document.write( "\"p%28x%29+=+-2%28x%2B3%29%28x%2B1%29%28x-4%29%28x%2B5%2F24%29\"

\n" ); document.write( "\"graph%28400%2C400%2C-5%2C5%2C-50%2C50%2C-2%28x%2B3%29%28x%2B1%29%28x-4%29%28x%2B5%2F24%29%29\"

\n" ); document.write( "Or we could do it with two additional linear factors, like (x+5/8) and (x+1/3):

\n" ); document.write( "\"p%28x%29+=+-2%28x%2B3%29%28x%2B1%29%28x-4%29%28x%2B5%2F8%29%28x%2B1%2F3%29\"

\n" ); document.write( "

\n" ); document.write( "I suspect, however, that it was not intended for you to do that kind of thing to answer the question. I think the statement of the problem simply put too many requirements on the function to make an answer possible.

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