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Inner product [tex]<f,g>=\int_{-1}^{1}f(x)g(x)dx[/tex].

Find a monic polynomial orthogonal to all polynomials of lower degrees.

Taking a polynomial of degree 3.

[tex]x^3+ax^2+bx+c[/tex]

Need to check [tex]\gamma, x+\alpha, x^2+\beta x+ \lambda[/tex]

[tex]\int_{-1}^{1}(\gamma x^3+\gamma a x^2 +\gamma bx + \gamma c)dx[/tex]

[tex]=\frac{\gamma x^4}{4}+\frac{\gamma a x^3}{3}+\frac{\gamma b x^2}{2}+\gamma c x|_{-1}^{1}[/tex]

[tex]=\frac{2\gamma a}{3}+2\gamma c=0\Rightarrow c=-\frac{a\gamma}{3}[/tex]

[tex]\int_{-1}^{1}(x^3+ax^2+bx+c)(x+\beta)dx[/tex]

[tex]\int_{-1}^{1}\left(x^4+ax^3+bx^2-\frac{a\alpha x}{3}+\beta x^3 +\alpha\beta x^2+b\beta x-\frac{a\alpha\beta}{3}\right)dx=6+10b+10a\beta-10a\alpha\beta=0[/tex]

What do I do with that?