document.write( "Question 577369: give an example of polynomials p and q of degree 3 such that p(1)=q(1), p(2)=q(2), p(3)=q(3), but p(4) =(not equal) q(4). Show that your polynomials satisfy these conditions. \n" ); document.write( "

Algebra.Com's Answer #370158 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
There is probably a very simple and elegant solution, but I will only see it after I post this messy one,
\n" ); document.write( "Consider the function $\"f%28x%29=p%28x%29-q%28x%29\"$
\n" ); document.write( " $\"f%281%29=p%281%29-q%281%29=0\"$
\n" ); document.write( " $\"f%282%29=p%282%29-q%282%29=0\"$
\n" ); document.write( " $\"f%283%29=p%283%29-q%283%29=0\"$
\n" ); document.write( "There are two many options, but I would try the simplest polynomial function with those three zeros:
\n" ); document.write( "

\n" ); document.write( "I can split $\"f%28x%29\"$ into $\"p%28x%29\"$ and $\"-q%28x%29\"$ many ways, but I need both polynomials to be of degree 3.
\n" ); document.write( "I'll try something. There are probably better, simpler ways.
\n" ); document.write( " $\"p%28x%29=2x%5E3-2x%5E2%2B2x-2=2%28x%5E3-x%5E2%2Bx-1%29\"$ and
\n" ); document.write( " $\"-q%28x%29=-x%5E3-4x%5E2%2B9x-4\"$ made up so that $\"p%28x%29%2B%28-q%28x%29%29=p%28x%29-q%28x%29=x%5E3-6x%5E2%2B11x-6\"$
\n" ); document.write( " $\"q%28x%29=x%5E3%2B4x%5E2-9x%2B4\"$
\n" ); document.write( "Polynomials p and q are of degree 3
\n" ); document.write( " $\"p%281%29=2%281%5E3-1%5E2%2B1-1%29=0\"$ and $\"q%281%29=1%5E3%2B4%2A1%5E2-9%2A1%2B4=1%2B4-9%2B4=0\"$ , so $\"p%281%29=q%281%29\"$
\n" ); document.write( " $\"p%282%29=2%282%5E3-2%5E2%2B2-1%29=2%288-4%2B2-1%29=2%2A5-10\"$ and $\"q%282%29=2%5E3%2B4%2A2%5E2-9%2A2%2B4=8%2B4%2A4-18%2B4=8%2B16-18%2B4=10\"$ , so $\"p%282%29=q%282%29\"$
\n" ); document.write( " $\"p%283%29=2%283%5E3-3%5E2%2B3-1%29=2%2827-9%2B3-1%29=2%2A20=40\"$ and $\"q%283%29=3%5E3%2B4%2A3%5E2-9%2A3%2B4=27%2B4%2A9-27%2B4=27%2B36-27%2B4=40\"$ , so $\"p%283%29=q%283%29\"$
\n" ); document.write( "I should not have to prove that, brcause there could only be 3 intersection points for two polynomials of degree 3, but ...
\n" ); document.write( " $\"p%284%29=2%284%5E3-4%5E2%2B4-1%29=2%2864-16%2B4-1%29=2%2A51=102\"$ and $\"q%284%29=4%5E3%2B4%2A4%5E2-9%2A4%2B4=64%2B64-36%2B4=128-36%2B4=96\"$ , so $\"p%284%29\"$ and $\"q%284%29\"$ are different. \n" ); document.write( "

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