Question

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

Answer 1

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