bovingal a ph respectively, and if p denotes the distance of a point P = (x, y) from the line l, then the locus of all points satisfying pip3 P4 = ap2P5 is given by аpгPs %3D %3D d (a +x)(a - x)(2a –...

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Extracted text: bovingal a ph respectively, and if p denotes the distance of a point P = (x, y) from the line l, then the locus of all points satisfying pip3 P4 = ap2P5 is given by аpгPs %3D %3D d (a +x)(a - x)(2a – x) = axy. diw bo This locus, which occurs in La Géométrie, was later called the Cartesian parabola, or trident, by Newton. bile 4. Show that the equation xr – x² + 2x +1= 0 has no positive roots. [Hint: Multiply by x + 1, which does not change the number of positive roots.] taird %3D 5. Find the number of positive roots of the equation x’ + 2x³ – x² + x – 1 = 0. %3D 6. From Descartes's rule of signs, conclude that the equation x2" – 1 = 0 has 2n – 2 imaginary roots. 7. Without actually obtaining these roots, show that (a) x³ +3x +7 = 0 and (b) x6 – 5x³ – 7x² + 8x + 20 = 0 %3D %3D both possess imaginary roots. 8. Verify the following assertions. If all the coefficients of an equation are positive and the equation involves no odd powers of r (a) then all its roots are imgi