Number Theory Questions

Question Number 163300 by amin96 last updated on 05/Jan/22

$$\int_{\mathrm{0}} ^{\mathrm{2}} \boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}\backsimeq\boldsymbol{{f}}\left(\boldsymbol{{a}}\right)+\boldsymbol{{f}}\left(\mathrm{2}−\boldsymbol{{a}}\right) \\$$$$\\$$For what values ​​of a is the following formula accurate for polynomials of degree 3?

Answered by MJS_new last updated on 05/Jan/22

$${f}\left({x}\right)={c}_{\mathrm{3}} {x}^{\mathrm{3}} +{c}_{\mathrm{2}} {x}^{\mathrm{2}} +{c}_{\mathrm{1}} {x}+{c}_{\mathrm{0}} \\$$$$\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}={f}\left({a}\right)+{f}\left(\mathrm{2}−{a}\right) \\$$$$\mathrm{4}{c}_{\mathrm{3}} +\frac{\mathrm{8}}{\mathrm{3}}{c}_{\mathrm{2}} +\mathrm{2}{c}_{\mathrm{1}} +\mathrm{2}{c}_{\mathrm{0}} =\mathrm{2}\left(\mathrm{3}{c}_{\mathrm{3}} +{c}_{\mathrm{2}} \right){a}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{3}{c}_{\mathrm{3}} +{c}_{\mathrm{2}} \right){a}+\mathrm{2}\left(\mathrm{4}{c}_{\mathrm{3}} +\mathrm{2}{c}_{\mathrm{2}} +{c}_{\mathrm{1}} +{c}_{\mathrm{0}} \right) \\$$$${a}^{\mathrm{2}} −\mathrm{2}{a}+\frac{\mathrm{2}}{\mathrm{3}}=\mathrm{0} \\$$$${a}=\mathrm{1}\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{3}} \\$$