Let P be the vector space of all polynomials (of all degrees) with real coefficients. In...

 

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Let P be the vector space of all polynomials (of all degrees) with real coefficients. In this problem, we will consider the linear functions d : P → P and s : P - P defined by d(p(x)) = p'(x), s(p(x)) = xp(x). In words, d is the function that takes the derivative of a polynomial, and s is the function that multiplies a polynomial by x. (a) Let p(x) = -x – 2x2 + x+3 and compute the following: d(p(x)) = -3x^2-4x+1 s(p(x)) = -x^4-2x^3+x^2+3x d(s(p(x))) = -4x^3-6x^2+2x+3 s(s(d(d(p(x))))) = -6x^3-4x^2 (b) Find a non-zero solution p(x) E P of the linear equation Let P be the vector space of all polynomials (of all degrees) with real coefficients. In this problem, we will consider the linear functions d : P → P and s : P - P defined by d(p(x)) = p'(x), s(p(x)) = xp(x). In words, d is the function that takes the derivative of a polynomial, and s is the function that multiplies a polynomial by x. (a) Let p(x) = -x – 2x2 + x+3 and compute the following: d(p(x)) = -3x^2-4x+1 s(p(x)) = -x^4-2x^3+x^2+3x d(s(p(x))) = -4x^3-6x^2+2x+3 s(s(d(d(p(x))))) = -6x^3-4x^2 (b) Find a non-zero solution p(x) E P of the linear equation