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Number of Real Roots.

and in this case the row of signs when x is

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so that the two different roots of the equation are, in this case, real.

The row of signs when. x 0 is

(like b),, (unlike b);

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so that one of the roots is positive and the other negative. Third case. When a is negative and of such a value that U" is positive or

(— }, a)3 > (1 b)?

in which case, the row of signs when x = ∞ is

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so that all three of the roots of the equation are real.

The row of signs when x = 0 is

(like b),, (like b), -.

If, then, b is positive the equation has one positive real root and two negative ones; and if b. is negative, it has two positive real roots and one negative one.

3. Find the number of real roots of the equation

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First case. If n is even, the row of signs, when x = ∞

is, then,

++, (unlike a);

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so that there is no real root when a is positive, and two real roots when a is negative, which agrees with art. 285.

is

Second case. If n is odd, the row of signs when x

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so that, in either case, there is only one real root, which is, by art. 284, of a sign unlike that of a.

4. Find the number of real roots of the equation

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U" — — a" (n — 1 ) n — 1 ———— nTM ( ———— b)n-1.

First case. When ʼn is even and greater than 2, and U" positive, that is, when b is positive, and

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so that when a is positive, there is no real root, and when

a is negative there are two real roots. In the latter case,

the row of signs when x = 0 is

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Number of Real Roots.

Second case. When n is even and greater than 2, and U" zero, that is when b is positive and

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so that in either case there is no other real root than the above equal root.

Third case. When n is even and greater than 2, and U" negative, that is,

n

(9)*>()"
(으)

n

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so that when a is negative there is no real root, and when a is positive there are two real roots. In the latter case, the row of signs when x = 0 is

(like b), +, ‡ (unlike b), —;

so that when b is positive, both the roots are negative, and when b is negative, one of the roots is positive and the other negative, which agrees with art. 287.

Fourth case. When n is odd, and U" positive, that is, when a is negative and

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Number of Real Roots.

in which case, the row of signs when x = ∞ is

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so that the equation has three real roots; the row of signs when x=0 is

(like b), -, (unlike b), +;

so that when b is negative, one of the real roots is positive and the other two negative; and when b is positive, one of the real roots is negative and the other two positive.

Fifth case. When n is odd, and U" zero, that is, when a is negative and

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so that there is another real root besides the above equal root. The row of signs when x = 0 is

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so that one of the roots is positive and the other negative.

Sixth case. When n is odd, and U" negative, that is,

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Number of Real Roots.

in which case, the row of signs when x = ∞ is

+, +, F (unlike a), — ;

when x

∞ it is

−, +, ± (like a),

so that there is only one real root, which, by art. 284, has a sign contrary to that of its last term.

4. Find the number of real roots of the equation

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Ans. It has one positive real root.

5. Find the number of real roots of the equation

x410 x3 35 x2

50x240.

Ans. It has four positive real roots.

6. Find the number of real roots of the equation

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Ans. It has three positive real roots and one negative

7. Find the number of real roots of the equation

x2 + 8 x2 + 16 x 440

-0.

one.

Ans. One positive root and one negative root.

293. Sturm's theorem is perfect in always giving the number of real roots, but often requires so much labor, that theorems, which are much less perfect, may be used with great advantage.

294. Stern's Theorem. Denote the first member of the equation

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by u and its first, second, &c. derivatives by U, U', &c.

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