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
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
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter12: Conic Sections
Section12.5: Rotation Of Axes
Problem 1E: Suppose the x- and y-axes are routed through an acute angle to produce the new X- and Y-axes. A...
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Number 4
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc27b03fb-27ff-4129-a29b-052f3c3c0c2f%2F938b0db5-0470-4e59-abdb-9052d7b4d9a5%2F2whgcvd.jpeg&w=3840&q=75)
Transcribed Image 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
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