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A possible HKAL Pure problem?

三月 5, 2009

Let A = \begin{pmatrix} 1 & x & x^3\\ 1 & y & y^3 \\1 & z & z^3 \end{pmatrix}

(a) By expanding the determinant along the last row (or otherwise), show that det(A) is a cubic polynomial in z, and we call det(A) = f(z). (1 point)

(b) Show that f(x) = f(y) = 0, and hence conclude that f(z) = (y - x)(z - x)(z - y)(z - r) for some r in terms of x and y. (5 points)

(c) Using the relationship between the roots and coefficients of a polynomial (or otherwise), show that r = -x - y. (3 points)

Posted in HKAL, Linear Algebra, Pure Mathematics, Teaching |

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