PolynomialsRecall that a polynomial of degree n is an expresion of the formNiMsJiomSSRhX25HNiIiIiIpSSJ4R0YmSSJuR0YmRidGJyomLUkjYV9HRiY2IywmRipGJ0YnISIiRicpRilGL0YnRic=... NiMsKComSSRhXzJHNiIiIiIqJEkieEdGJiIiI0YnRicqJkkkYV8xR0YmRidGKUYnRidJJGFfMEdGJkYn where a_n is not zero.Find a polynomial that passes through the points (1,6), (2,4).First way: Using the quation of the linem:=(6-4)/(1-2);

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y-6=m*(x-1);

NiMvLCZJInlHNiIiIiIhIidGJywmSSJ4R0YmISIjIiIjRic=

Second way: Start with y=m*x+b and find m and b solving a system of equations.m:='m';

NiM+SSJtRzYiRiQ=

solve({6=m+b,4=m*2+b},{m,b});

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Third way: use the package CurveFittingwith(CurveFitting):PolynomialInterpolation([[1,6],[2,4]],z);

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We know how to find the polynomial passing through two points.Plot the points (1,-1), (2,-2) and (3,-3). Find a polynomial that passes through all of them.data0:=[[1,-1],[2,-2],[3,-3]];

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plot(data0,x=-5..5,style=point);

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Using the fact that the three points are on a line, one can proceed as in the first case.PolynomialInterpolation(data0,x);

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Find a polynomial that passes through the points (1,6), (2,4) and (3,5)data1:=[[1,6],[2,4],[3,5]];

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plot(data1,x=0..4,style=point,symbol=box);

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p:=x->a*x^2+b*x+c;

NiM+SSJwRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJkkiYUdGJSIiIjkkIiIjRi8qJkkiYkdGJUYvRjBGL0YvSSJjR0YlRi9GJUYlRiU=

solve({p(1)=6,p(2)=4,p(3)=5},{a,b,c});

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assign(%);a;

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plots[display](plot(data1,x=0..4,style=point,symbol=box),plot(p(x),x=0..4));

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PolynomialInterpolation(data1,x);

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Let's try with the polynomial of degree 2, a*x^2+b*x+cEXERCISE: What happens if you try to find a polynomial of degree 3 passing through the points (1,6), (2,4) and (3,5)?Now, consider the following data. restart;data:=[seq([i,i/2],i=1..10)];

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Plot the points of data and find a polynomial passing through all of them.plot(data,style=point);

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Consider now the following data. Plot data and data2 together. data2:=[seq([i,i/2],i=1..7),[8,3.4],[9,4.7],[10,5]];

NiM+SSZkYXRhMkc2IjcsNyQiIiIjRigiIiM3JEYqRig3JCIiJCNGLUYqNyQiIiVGKjckIiImI0YyRio3JCIiJ0YtNyQiIigjRjdGKjckIiIpJCIjTSEiIjckIiIqJCIjWkY9NyQiIzVGMg==

plot({data, data2},style=point, symbol=[box,diamond],color=[red,blue]);

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Find a polynomial passing through the points of data2.with(CurveFitting):P:=PolynomialInterpolation(data2,x);

NiM+SSJQRzYiLDYqJEkieEdGJSIiKiQhKyxGVFtrISM5KiRGKCIiKSQiK19PekRJISM3JCIrLSsrK0ghIikiIiIqJEYoIiIoJCErJVFfJW9nISM2RigkIStAZEclMylGNCokRigiIickIitybTsvbyEjNSokRigiIiMkIitpIz1LPSpGNCokRigiIiYkISsudiQ0byUhIioqJEYoIiIkJCErUmNhImUmRjQqJEYoIiIlJCIrLHZWUT9GNA==

Plot the polynomial of data adn the polynomial of data2 in the same graph. Compare the results.plot({P(x),x/2},x=0..11,y=0..10,thickness=4);

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

Exercise: Find a polynomial r(x) that interpolates the data (1,-5), (2,-8.3),(3,-11),(4,-14) and (5,-17) by defining first and structure of points and then using PolynomialInterpolation. Plot the results.data3:=[[1,-5],[2,-8.5],[3,-10.8],[4,-14.21],[5,-17]];

Repeat the above procedure with the data [[1,-5],[2,-8],[3,-11],[4,-14]]. Compare both results by making the graph of both functions together. Find a value of x for which the value of the two functions differs significatively.data4:=[[1,-5],[2,-8],[3,-11],[4,-14]];

s:=PolynomialInterpolation(data4,x);

plots[display](plot(data3,style=point,color=blue),plot({r,s},x=0..6));