In all other respects, the operation is entirely similar to division in Arithmetic. Since all other similar cases may be treated in the same way, we have the following

RULE.

I. Arrange both polynomials with reference to the same letter.

II. Divide the first term of the dividend by the first term of the divisor, for the first term of the quotient. Multiply the divisor by this term of the quotient, and subtract the product from the dividend.

III. Divide the first term of the remainder by the first term of the divisor, for the second term of the quotient. Multiply the divisor by this term, and subtract the product from the first remainder, and so on.

IV. Continue the operation, until a remainder is found equal to 0, or one whose first term is not divisible by that of the divisor.

When a remainder is found equal to 0, the division is exact. When a remainder is found whose first term is not divisible by that of the divisor, without giving rise to fractions, the exact division is impossible. In that case, write the last remainder after the quotient found, placing under it the divisor, in the form of a fraction.

Here the quotient is fractional, and the division is not

In this example, the operation does not terminate, but may be continued to any desired extent.

EXAMPLES.

1. Divide a2 + 4ax + 4x2 by a + 2x.

Ans. a + 2x.

3. Divide a3 + 5a2x + 5αx2 + x3 by a + x.

Ans. a2+ 4αx + x2.

4. Divide a 4a3y+6a2y2 - 4ay3 + y by a2-2ay

Ans. a b

5. Divide a-b1 by a3+a2b+ab2+b3.

8. Divide 2n + x2ny2n + yn by x2+x"y" + y2n.

9. Divide a2 — b2 + 2bc c2 by a · b + c.

16. Divide a3 + a3b2 + 2a2b3 — b5 by a2 — ab + b2.

Ans. a3 + a2b + ab2 + 2h ́ +

17. Divide 3 + ax2 + bx + c by x

Ans. x2 + rx + ax + p2 + ar + b +

r.

23 + ar2 + br + c

x-r

18. Divide 1+ 2x by Ans.

1— 3x.

15x + 15x2 + 45x3 +, &c

35. A FORMULA, is an algebraic expression of a general rule, or principle.

The verification of the following formulas, affords additional exercises in Multiplication.

36. Formulas serve to shorten algebraic operations, and are also of much use in the operation of factoring. When translated into common language, they give rise to practical rules. These three may be translated as follows, since x and y are any quantities whatever :

1. The square of the sum of any two quantities, is equal to the square of the first, plus twice the product of first and second, plus the square of the second.

2. The square of the difference of any two quanti ties, is equal to the square of the first, minus twice the product of the first and second, plus the square of the second.

3. The product of the sum and difference of two quantities, is equal to the square of the first, minus the square of the second.

first rule,

EXAMPLES.

1. Let it be required to find the square of 2a + 3.x. The square of 2a, is 4a2; twice the product of 2a and 3x, is 12ax; the square of 3x, is 9x2.

Hence, by the

3. Find the product of 2a + 3x, and 2a - 3x. By the third rule, we have, as before,

= 4a2 - 12ax +9x2.

In like manner, solve the following examples:

4. Find the square of ax + by.