Factoring and Algebra - A Selection of Classic Mathematical Articles Containing Examples and Exercises on the Subject of Algebra (Mathematics Series)

By Read Books Ltd.

This book contains classic material dating back to the 1900s and before. The content has been carefully selected for its interest and relevance to a modern audience. Carefully selecting the best articles from our collection we have compiled a series of historical and informative publications on the subject of mathematics. The titles in this range include "Ratio and Proportion" "Simple Equations" "Simultaneous Equations" and many more. Each publication has been professionally curated and includes all details on the original source material. This particular instalment, "Factoring and Algebra" contains a selection of classic educational articles containing examples and exercises on the subject of algebra. It is intended to illustrate aspects of factoring and serves as a guide for anyone wishing to obtain a general knowledge of the subject. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

• Language: English
• Publisher: Read Books Ltd.
• Release date: Oct 25, 2016
• ISBN: 9781473358775

Book preview

FACTORING.

62. FACTORING is the resolving a quantity into its factors.

63. The factors of a quantity are those integral quantities whose continued product is the quantity.

NOTE.—In using the word factor we shall exclude unity.

64. A PRIME QUANTITY is one that is divisible without remainder by no integral quantity except itself and unity.

Two quantities are mutually prime when they have no common factor.

65. The PRIME FACTORS of a quantity are those prime quantities whose continued product is the quantity.

66. The factors of a purely algebraic monomial quantity are apparent. Thus, the factors of a² b x y z are a × a × b × x × y × z.

67. Polynomials are factored by inspection, in accordance with the principles of division and the theorems of the preceding section.

CASE I.

68. When all the terms have a common factor.

1. Find the factors of a x — a b + a c.

OPERATION.

(a x — a b + a c) = a (x — b + c)

As a is a factor of each term it must be a factor of the polynomial; and if we divide the polynomial by a, we obtain the other factor. Hence,

RULE.

Write the quotient of the polynomial divided by the common factor in a parenthesis, with the common factor prefixed as a coefficient.

2. Find the factors of 6 x y — 72 x y² + 18 a x² y³.

Ans. 6 x y (1 — 12 y + 3 a x y²).

NOTE.—Any factor common to all the terms can be taken as well as 6 x y; 2, 3, x, y, or the product of any two or more of these quantities, according to the result which is desired. In the examples given, let the greatest monomial factor be taken.

3. Find the factors of x + x².   Ans. x (1 + x).

4. Find the factors of 8 a² x² + 12 a³ x⁴ — 4 a x y.

Ans. 4 a x (2 a x + 3 a² x³ — y).

5. Find the factors of 5 x⁴ y² + 25 a x⁵ — 15 x³ y³.

Ans. 5 x³ (x y²+ 5 a x² — 3 y³).

6. Find the factors of 7 a x — 8 b y + 14 x².

7. Find the factors of 4 x² y² — 28 x³ y⁴ — 44 x⁴ y².

8. Find the factors of 55 a² c — 11 a c + 33 a² c x.

9. Find the factors of 98 a² x² — 294 a³ x² y².

10. Find the factors of 15 a² b² c d — 9 a b² d² + 18 a³ c² d⁴.

CASE II.

69. When two terms of a trinomial are perfect squares and positive, and the third term is equal to twice the product of their square roots.

1. Find the factors of a² + 2 a b + b².

OPERATION.

a² + 2 a b + b² = (a + b) (a + b)

We resolve this into its factors at once by the converse of the principle in Theorem II. Art. 58

2. Find the factors of a² — 2 a b + b².

OPERATION.

a² — 2 a b + b² = (a — b) (a — b)

We resolve this into its factors at once by the converse of the principle in Theorem III. Art. 59. Hence,

RULE.

Omitting the term that is equal to twice the product of the square roots of the other two, take for each factor the square root of each of the other two connected by the sign of the term omitted.