354. If an ordinary train leave the terminus at 4 P. M. at the rate of ten miles an hour, and increase its speed at the rate of two miles an hour every ten minutes; and an express train leave the same terminus at 4 hrs. 10 min. P. M. starting at the same speed as the other, but increasing its speed every ten minutes, in the ratio of 5 to 7; what would be the relative position of these trains at the end of fifty minutes after the starting of the express?

355. Find the number by which the numbers 20, 50, and 100 being severally increased, the results may be

(1) In geometrical progression :

(2) In harmonical progression.

356. Shew that three different quantities cannot be in harmonic progression and also in arithmetic progression; but that, if they could, they must also be in geometric progression.

357. If a, b, c be in harmonic progression, show that

358. Show that if a polynomial in z be divided by z- - v, the final remainder will be the same as the original polynomial, only having v in the place of z.

359. Apply the property stated in the last question to the determination of the value of the expression

v3 — 5v1 — 3v3 — 5v + 2,

when v=

- ·5.

360. Find, by synthetic division, the values of the expression w*- 3w3 - 12w3 + w — 53,

when w=3, w = 6, w='58, and w=- -2.73.

361. State the principle upon which the method of determining the value of an algebraical expression by synthetic division depends, and apply it to the determination of the value of

y* − 4y3 + 9y + 3, when y = -3.

362. Find, by synthetic division, the values of the following fractions, when x=3:

363. Find the value of x3- 7x3+5x+8; when x=-1.2.

367. Find the value of x3 – 17x3 + 2x2 + 20, when x = 4; by division.

368.

Show that if any equation be divisible without remainder by x-a, a is a root.

369. Show that every equation has the same number of roots as there are units in the highest exponent of the unknown quantity in it.

3, 2 + 3 √(−1),

370. Form the equation whose roots are 2-3(-1) and 5, without multiplication, and prove its correctness by multiplication.

371. Find the middle term of the equation whose roots are 1, – 2, 3 and −4; without determining any other term.

372. Determine by inspection the roots of the equation

ax3 − (b + ac − a3d) x2 + (bc — abd — a2cd) x + abcd = 0.

373. Determine the last term but one of the equation whose roots are 1, −2, 3, − 4, 5 and – 6, without determining any other

term.

374. Of how many products is the coefficient of the middle term of an equation having 8 roots, the sum ?

375. Form the equation whose roots are 2, 5, and 4±√(−7).

376. Find the last term but one of the equation whose roots are 1, 2, -3, 4 and 5.

377. The second term of an equation is 9x, its last term is 2520, and three of its roots are 5, -6 and −7; find the remaining roots.

378. Two roots of the equation

x*- 35x2 + 90x – 56 = 0

are 1 and 2; find the remaining roots.

379. The roots of the equation

x2-6x2-4x + 24 = 0

are in arithmetical progression: find them.

380. Show that if the roots of the equation
ax + bx2 + cx2 + dx + e = 0

381. Two roots of an equation whose second and last terms are respectively 112 and +30, are 1 and 5; what are its other roots?

382. Determine the last term but three of the equation whose roots are 1, 2, 3, − 1, − 2, and 3; without finding any other term.

383. Assuming that if an equation in x be divisible by x − r, r is a root of it; state how it appears that if p, q, s, t, v, &c. be also roots of such an equation, the successive quotients arising from the division of the original equation by any number of the binomials x-p, x— q, x−r, x − 8, &c. will, when equated to 0, be equations whose roots are, in each case, the second terms of the binomials by which it has not been divided.

384.

385,

t

1

Find all the roots of the equation

x3- 2x - 10x3 + 20x2 + 9x - 180.

Determine whether any of the digits less than 6 are

roots of the equation

x3 – 6×3 + 9×1 + 2x3 — 13x2 + 26x − 15 = 0.

386. Find all the roots of the equation

387.

For what value of n will the roots of the equation 2x2 + 8x+n=0 be equal?

388. Show that if an equation have p roots each equal to ", and q roots each equal to s, the limiting equation will have p-1 roots each equal to r, and q-1 each equal to s: and state the law of the formation of the coefficients of the limiting equation from those of the original.

389. Supposing

(y-p). (y-q). (y − r)°. (y — 8)3. (y-t). (y-v)

to be the form into which the greatest common measure between an equation and its limiting equation can be transformed; state

what you would infer therefrom with regard to the roots of that equation.

and, by means of them, complete its solution.

392.

Determine the equal roots of the equation

x2 - 16x3+90x2 - 208x + 169 = 0.

393. By means of its equal roots solve the equation

x3- x2-16x - 20 = 0.

394. Solve the equation

x2-6x+15x2-18x+9= 0,

which has equal roots.

395. Determine the equal roots of the equation

x5+5x-5x3-45x2 + 108 = 0.

396. By means of its equal roots, solve the equation

have one root common, find it; and thence determine the remaining roots of both.

398. Prove that if the alternate signs of an equation be changed, the roots of the new equation will be the same as those of the original with contrary signs.

399. Prove that if we wish to form, from any equation, another, whose roots shall be greater, or less, by a given quantity