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

10. Divide 24—6x2y2 (16xy3-15y4 by x2 + 2xy

+ 3y2.

11. Divide ax3 a2x2 + bx2 + b2 by ax

b.

nqx + nr

Ans. px2 + qx

2a-8x5 + 17a-4x6 5х7 24a4x8

by

16. Divide a + a3b2 + Qa2b3 — b5 by a2 — ab + b2.

V. USEFUL FORMULAS.

Definitions.

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

Formulas are used to shorten algebraic operations, such as the formation of powers, factoring, and the like.

Illustration.

36. Let the following formulas be verified by actual multiplication:

If we suppose x and y to represent any two quantities, and then translate these formulas into words, we have the following principles:

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

2o. The square of the difference of any two quantities, 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 any two quantities, is equal to the square of the first, minus the square of the second.

The method of applying these principles is shown in the following

EXAMPLES.

1. Let it be required to find the square of 2a + 3x.

The square of 2a, is 4a2; twice the product of 2a and 3x, is 12ax; the square of 3x, is 9x2.

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

(2a + 3x) (2a-3x) = 4a2 9x2.

In like manner let the following operations be performed:

4. Find the square of ax + by.

10. Find the product of 2a + 3x, and 2a - 3x.

Ans. 4a2 9x2.

11. Find the product of 7b+ 4c, and 7b4c. Ans. 4962

12. Find the product of 8xy + 3x2, and 8xy

16c2.

3x2.

Ans. 64x2y2-9x1.

By reversing these operations, the squares and products above found may be resolved into binomial factors.

The following additional formulas may be verified by actual multiplication, or division, with the exception of the ninth and tenth. The demonstration of these will be given in the appendix.

4°. (x + a) (x + b) = x2 + (a + b)x + ab.

5°. (x2 + xy + y2) (x − y) = x3 — y3.

6°. (x2 — xy + y2) (x + y) = x2 + y3.

ry°. (x2 + xy + y2) (x2 — xy + y2) = x2 + x2y2 + y1. 8°. (x + y) (x − y) (x2 + y2) = x4 — y1.

ymn

yn

= xmn−n + xmn-2nyn + xmn-3ny2n + &c.

37. Factoring is the operation of separating, or resolving, a quantity into factors.

No general rule can be given for factoring: in most cases the operation is performed by inspection and trial. The methods of

proceeding are best illustrated by examples.

Methods of Factoring.

38. If every term of a polynomial contains the same monomial factor, that factor is one factor of the polynomial, and the other factor is equal to the quotient of the polynomial by the monomial factor.

EXAMPLES.

1. Factor the polynomial Sa2x2 + 4a3x.

Here, we see that 4a2x is a factor common to each term, hence it is one of the required factors. Dividing by 4ɑ2x, we have the quotient, 2x+a, which is the other factor; or,

In like manner, the following polynomials may be factored.