3. Given the logarithmic tangents of 17°, 18°, 19°, 20°; viz. 485339, 511776, 536972, 561066, to find the tangent of 18°. Ans. 524520.

4. Given the natural sines of 24°, 25°, 26°, 27°; viz. 40674, 42262, 43837, 45399, to find the natural sine of 25°%.

ART. 535. EQUATIONS of any degree may be produced from simple equations, by multiplication. The manner in which they are compounded will be best understood, by taking them in that state in which they are all brought on one side by transposition. (Art. 183.) It will also be necessary to assign, to the same letter, different values, in the different simple equations.

Suppose, that in one equation, x=2
And, that in another,

By transposition,

x=3

x-2=0

And mult. as before, x4-14x3 +71x2-154x+120=0, &c.

Collecting together the products, we have

(x-2)(x-3)

(x-2)(x-3)(x-4)

=x-5+6=0

=x3-9x2+26x-24=0

(x-2)(x-3)(x-4)(x-5)=x+-14x3 +71x2-154x+120=0, &c.

That is, the product

of two simple equations is a quadratic equation;
of three simple equations, is a cubic equation;

of four simple equations, is a biquadratic, or an equation of the fourth degree, &c. (Art. 317.)

Or a cubic equation may be considered as the product of a quadratic and a simple equation; a biquadratic, as the product of two quadratic; or of a cubic and a simple equation, &c.

In each case, the exponent of the unknown quantity, in the first term, is equal to the degree of the equation; and, in the succeeding terms, it decreases regularly by 1, like the exponent of the leading quantity in the power of a binomial. (Art. 485.) In a quadratic equation, the exponents are 2, 1, 3, 2, 1, 4, 3, 2, 1, &c.

In a cubic equation,
In a biquadratic,

The number of terms, is greater by 1, than the degree of the equation, or the number of simple equations from which it is produced. For besides the terms which contain the different powers of the unknown quantity, there is one which consists of known quantities only. The equation is here supposed to be complete. But if there are, in the partial products, terms which balance each other, these may disappear in the result. (Art. 104.)

536. Each of the values of the unknown quantity is called a root of the equation.

Thus, in the example above,

The roots of the quadratic equation are

of the cubic equation,

of the biquadratic,

3, 2,

4, 3, 2,

5, 4, 3, 2.

The term root is not to be understood in the same sense here, as in the preceding sections. The root of an equation 's not a quantity which multiplied into itself will produce the equation. It is one of the values of the unknown quantity; and when its sign is changed by transposition, it is a term in one of the binomial factors which enter into the composition of the equation of which it is a root.

The value of the unknown letter x, in the equation, is a quantity which may be substituted for x, without affecting the equality of the members. In the equations which we are now considering, each member is equal to 0; and the first is the product of several factors. This product will continue to be equal to 0, as long as any one of its factors is 0. (Art. 106.) If then in the equation

(x-2)x(x-3)× (x−4) × (x-5)=0,

we substitute 2 for x, in the first factor, we have

0×(x−3)×(x−4)×(x-5)=0.

So, if we substitute 3 for x, in the second factor, or 4 in the third, or 5 in the fourth, the whole product will still be 0. This will also be the case, when the product is formed by an actual multiplication of the several factors into each other.

Thus, as
So

And

x3-9x2+26x-24=0; (Art. 535.)
23-9×22+26×2-24=0,

33-9x32+26×3-24=0, &c.

Either of these values of x, therefore, will satisfy the conditions of the equation.

We have thus far been considering higher equations as formed, by multiplication, from simple equations. But the inquiry may arise, whether every equation of a higher degree can be regarded as the product of two or more simple equations. It is proposed, in the following articles, to answer this inquiry, and to bring into view a number of the most important general properties of equations.

537. An equation of the mth degree consists of ", the several inferior powers of x with their co-efficients, and one term in which x is not contained. If A, B, C,....T, be put for the several co-efficients, and U for the last term, then

xTM+AxTM−1+Вxm-2+Сxm-3....+Tx+U=0,

will be a general expression for an equation of any degree. Any quantity which, substituted for x, will make the members equal, is called a root of the equation. (Arts. 335, 536.) If (a) is a root of the general equation of the (m)th degree, the first member is exactly divisible by (x—a).

For by substituting a for x, we have

am+Aam-1+Bam-3+Cam-3....+Ta+U=0.

And transposing terms,

U-am-Aam-1-Bam-2 — Cam-3

Substituting this value for U, in the original equation,

x+Аxm-1+Вxm-2+Сxm-3

- am — Aam-1 - Bam-2 - Cam-3

+ Tx

=0.

- Ta

....

Or, uniting the corresponding terms,

(x-a)+4(x-1-am-1)+B(x-2-am-2)+

C(xm-3-am-3)....+T(x-a)=0.

-

In this expression, each of the quantities (xTM — aTM), A(x-1-am-1), &c. is divisible by x-a; (Art. 130:) therefore the whole is divisible by x-a.

Conversely, If the first member of any equation be divisible by x-a, a is a root of the equation. For (Art. 114,) this member may be resolved into two factors, of which x-a is one; and (Art. 106,) it must itself become zero, when x=a, because the factor x-a becomes zero. The equation will therefore be satisfied, by giving to x the value a; and a must be one of its roots.

Ex. 1. Prove that 2 is a root of the equation

x3-7x2+12x-4=0.

This is to be done, by dividing by x-2. If there is no remainder, 2 must be a root of the equation.

Ex. 2. Prove that 3 is not a root of the equation

x3+2x2-6x-4=0.

When we divide here by x-3, we find a remainder; which shows that 3 is not a root.

Ex. 3. Find whether -1 is a root of the equation

Ex. 4. Find whether 1 is a root of the equation

x4-2x3-11x2-8x+15=0.

Ex. 5. Find whether -5 is a root of the equation

x5+3x3-64x2+x+23=0.

538. Every equation of the (m)th degree has exactly (m) roots.

It will be assumed that every equation has at least one root. Let a be a root of the equation

x+Ax-1+Bx-2....+Tx+U=0.

The first member is divisible by (x-a) (Art. 537.) If we divide by this, the quotient will be a polynomial of the degree (m-1); which may be written thus,

xm-1+A'x-3+B'xm-3 &c.

If we make this equal to zero, and suppose b to be a roor of the equation thus formed, the first member will be divisible by x-b; and if we divide by this, the result will be a polynomial of the degree (m-2). By proceeding in this way till we have divided (m-1) times, we shall obtain a

simple equation, having only one root, which may be denoted by i. Hence the original equation has m roots, a, b, c,...l; and the first member of it is composed of the m factors, x-a, x-b, x-c,...x-l.

The equation can have no other root; for none of the factors x-a, x-b, &c. of which the first member is composed, can be zero, unless x equals one of the quantities, a, b, &c.

Ex. 1. One root of the equation

x3-4x2+2x+3=0

is 3. What are the other roots?

If we divide by x-3, we obtain the equation x2−x−1=0; which must contain the two required roots. By solving this quadratic, the roots will be found to be

Ex. 2. One root of the equation

x3-5x+3x+9=0

2

are 1 and 2. What are the other roots? Ans. −1±√−5.

Ex. 5. Two roots of the equation

are 1 and -3.

x2+x3-7x2 −x+6=0

What are the other roots?

Ex. 6. One root of the equation

x1 −2x3+x2 -5x-9=0

is 1. Find the equation containing the other roots.

539. The roots of an equation are not always real. Some or even all of them may be imaginary. In the fourth example above, two of the roots are imaginary, and two real.

It often happens that the roots of an equation are not all unequal. Thus, in the second of the preceding examples, the roots are 3, 3, and -1; two of which are alike.