[alogo] Polynomial property

Consider a polynomial function with real coefficients F(x) = anxn + ... + a1x + a0. Draw a line y=bx+c and locate the intersection points (x1,y1), ..., (xn,yn) of the line with the graph (some of them may be complex). For all 1< k <= n , the following generalizations of Newton's relations are valid.

[0_0] [0_1] [0_2] [0_3] [0_4]
[1_0] [1_1] [1_2] [1_3] [1_4]
[2_0] [2_1] [2_2] [2_3] [2_4]
[3_0] [3_1] [3_2] [3_3] [3_4]

The proof follows easily from Newton's relations and the observation that G(x) = F(x) - (bx+c) is a polynomial with the same coefficients as F(x) except the two of lowest degree.
The picture above gives an application of this property for the case of a polynomial F(x) = a4x4+a3x3+a2x2+a1x+a0, of degree 4 of the above shape (having three local extrema, or equivalently having derivative with three real roots). The tangent PQ has B' (lying over B(-a3/(4a4)) as its middle. It is also parallel to the bitangent P'Q' of the graph. The location of B between the projections of P', Q' on the x-axis is also implied from the property. More is true for this particular configuration. All polynomial curves of degree four (i.e. curves represented by graphs of polynomials of degree four with real coefficients) are afine-axis-symmetric. The axis of symmetry is the vertical line at x = -a3/(4a4) and the direction P'Q' along which the symmetry is the bittangent direction. In the graph above all lines parallel to P'Q' intersect the graph in pairs of points symmetric with respect to their intersection point with the axis BB'.

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