Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. (1,2),(2,1),(3,3),(6,1)(1,2),(2,1),(3,3),(6,1).

Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. (1,2),(2,1),(3,3),(6,1)(1,2),(2,1),(3,3),(6,1).

Question
Polynomial graphs
asked 2020-11-09
Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. \(\displaystyle{\left({1},{2}\right)},{\left({2},{1}\right)},{\left({3},{3}\right)},{\left({6},{1}\right)}{\left({1},{2}\right)},{\left({2},{1}\right)},{\left({3},{3}\right)},{\left({6},{1}\right)}\).

Answers (1)

2020-11-10
Step 1
With PSKx_{1}=1, x_{2}=2, x_{3}=3\ and\ x_{4}=6, the Langrange interpolating polynomials give
\(\displaystyle{p}_{{{1}}}{\left({x}\right)}={\frac{{{\left({x}-{x}_{{{2}}}\right)}{\left({x}-{x}_{{{3}}}\right)}{\left({x}-{x}_{{{4}}}\right)}}}{{{\left({x}_{{{1}}}-{x}_{{{2}}}\right)}{\left({x}_{{{1}}}-{x}_{{{3}}}\right)}{\left({x}_{{{1}}}-{x}_{{{4}}}\right)}}}}\)
\(\displaystyle={\frac{{{\left({x}-{2}\right)}{\left({x}-{3}\right)}{\left({x}-{6}\right)}}}{{{\left({1}-{2}\right)}{\left({1}-{3}\right)}{\left({1}-{6}\right)}}}}\)
\(\displaystyle={\frac{{{\left({x}^{{{2}}}-{5}{x}+{6}\right)}{\left({x}-{6}\right)}}}{{{\left(-{1}\right)}{\left(-{2}\right)}{\left(-{5}\right)}}}}\)
\(\displaystyle=-{\frac{{{x}^{{{3}}}-{6}{x}^{{{2}}}-{5}{x}^{{{2}}}+{30}{x}+{6}{x}-{36}}}{{{10}}}}\)
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