Let px a0 a1x a2x2 and qx b0 b1x b2x2 be polynomials i

Let p(x) = a_0 + a_1x + a_2x^2, and q(x) = b_0 + b_1x + b_2x^2 be polynomials in P_2(R). Show that the function (p(x),q(x)) = p(-1)q(-1) + p(0)q(0) + p(1)q(1) defines an inner product on P_2(R). Compute (p(x),q(x)), ||p(x)||, and ||q(x)|| where p(x) = -1 + 2x + 3x^2 and q(x) m -4x + 2x^2 p(x) = - 1 + 2x - z^2 and q(x) = -8 - 3x + 2z^2

Solution

b) i) Given that p(x) = -1+2x + 3x² and q(x) = -4x+ 2x² = 0-4x+ 2x²

<p(x),q(x)> = -1.0 + 2x.-4x + 3x².2x² = -8x²+6x⁴

IIp(x)II =   [<p(x),p(x)>]^1/2 = [-1.-1+2x.2x+ 3x².3x²]^1/2 = [1+4x²+9x⁴]^1/2

    IIq(x)II =   [<q(x),q(x)>]^1/2 = [-4x.-4x+ 2x².2x²]^1/2 = [16x²+4x⁴]^1/2

ii) Given that p(x) = -1+2x -x²  and q(x) = -8-3x+ 2x²

<p(x),q(x)> = -1.-8 + 2x.-3x + -x².2x² = 8-6x²-2x⁴

IIp(x)II =   [<p(x),p(x)>]^1/2 = [-1.-1+2x.2x+ -x².-x²]^1/2 = [1+4x²+x⁴]^1/2

    IIq(x)II =   [<q(x),q(x)>]^1/2 = [-8.8+-3x.-3x+ 2x².2x²]^1/2 = [64+9x²+4x⁴]^1/2