example 973 the monic legendre polynomial of degree 2 is z2 3 s0 its zeros are theorem 972 guara

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Example 9.7.3. The monic Legendre polynomial of degree 2 is z^2 - 3. Its zeros are Â±âˆš3. Theorem 9.7.2 guarantees that âˆ«p(z) dx = W0p + W1pâˆš3 for all p(c) âˆˆ â„[x; 3], with W0 = âˆ«L0(c) dz. A straightforward calculation shows that L0(z) = âˆš(1-z^2) and L1(z) = 8(âˆš3+z) and integrating these gives W0 = W1 = 1. This implies that âˆ«(az^3 + bz^2 + cz + d) dz = aâˆ«(z^3) dz + bâˆ«(z^2) dz + câˆ«(z) dz + dâˆ«(1) dz = a(âˆš3) + b(âˆš3)^2 + c(âˆš3) + 4d + a(âˆš3) + b(âˆš3)^2 + c(âˆš3) + 4d = 2bâˆš3 + 2d for all a, b, c, d âˆˆ â„. The coefficients of terms of odd degree do not contribute in the computation because all odd functions integrate to 0 on [-1,1].

          Example 9.7.3. The monic Legendre polynomial of degree 2 is z^2 - 3. Its zeros are Â±âˆš3. Theorem 9.7.2 guarantees that
âˆ«p(z) dx = W0p + W1pâˆš3
for all p(c) âˆˆ â„[x; 3], with W0 = âˆ«L0(c) dz. A straightforward calculation shows that L0(z) = âˆš(1-z^2) and L1(z) = 8(âˆš3+z) and integrating these gives W0 = W1 = 1. This implies that
âˆ«(az^3 + bz^2 + cz + d) dz = aâˆ«(z^3) dz + bâˆ«(z^2) dz + câˆ«(z) dz + dâˆ«(1) dz = a(âˆš3) + b(âˆš3)^2 + c(âˆš3) + 4d + a(âˆš3) + b(âˆš3)^2 + c(âˆš3) + 4d = 2bâˆš3 + 2d
for all a, b, c, d âˆˆ â„. The coefficients of terms of odd degree do not contribute in the computation because all odd functions integrate to 0 on [-1,1].

example 973 the monic legendre polynomial of degree 2 is z2 3 s0 its zeros are theorem 972 guarantees that pz dx wo p w1p v3 for all pc rjx 3 with wi fl1lc dz a straightforward calculation 64373

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