x−(− a), or x + a is a factor.

1.

p-4.

m

5.

68. If the polynomial becomes 0 when a is put for x, then

Example: x + 2 is a factor of x8 + x2 - 2 x because

(-2)+(-2)2 -2(-2)=-8 + 4 + 4 = 0.

WRITTEN EXERCISES

By use of the Factor Theorem, prove that each polynomial has the factor named:

7.9x5+ (3a2-12 a)x-4a31⁄23 +3 a2x + a1.

X

a2.

In each polynomial, substitute the values given for x, and use the Factor Theorem to find the possible factors:

11. Apply the Factor Theorem to (a”±b′′) ÷ (a±b) and prove the results of Secs. 65 and 66.

REVIEW

WRITTEN EXERCISES

Perform the indicated operations, expressing fractional results in lowest terms:

1.

2a-3b4a+11c-2d+6-8a-9b+3c-4-5d

+2a-b.

2. x2y + 3 xy2+4 x2 − 2 xy − 3 x2y + 2 x2y2 — 4 xy +8 xy2

-

-7 y3-3x2y.

3. 5x-7y-(3x+4 y − 2) + 3 + (8 x − 7) − (2 x −8 y +13) +8(2x−1).

4. 4x-2c-(3x+2y+5c).

5. 5ac-4b+2 c[3 a − 5 b − (6 a − 2 b − 7c+3)] } .

If L=4x+2y-3, M=x-9y+1, R=y-5x, find:

6. L+M+R.

8. LM-3 R.

10. (L+M)2.

11. 3 L2-5 MR.

7. 3 L-2M+ R. 9. L2 M2.

If X=7a+2b, Y=2a-9b+3, Z=4b-7a-3, find:

Divide by detached coefficients :

35. 1632x + 24 x2 - 8 x3 + x4 ÷ (2-x).

36. 16 m1-32 m3 + 24 m2 - 8 m +1 ÷ (2 m −1).

·

—

39. a3- ab2+a2-ab. (Substitute b for a.)

SUMMARY

1. State the laws that govern the processes of addition for algebraic numbers.

Secs. 3, 9.

2. State the laws that govern the processes of multiplication for algebraic numbers.

Secs. 26, 32.

3. State and illustrate two extensions of the number system of algebra.

Secs. 16, 45.

4. Name eight type expressions used in factoring. Sec. 62. 5. State the Factor Theorem.

Sec. 67.

HISTORICAL NOTE

We have seen that algebra embraces the enlargement of the number field of arithmetic to include negative number and the extension of the basic processes to those numbers. Thus, the equation x + 3 = 1, which Ahmes could not solve and which was regarded as negligible until the sixteenth century, is no more exceptional than x+1=3, since algebra defines 13 to be 2. In a similar way the field of number was enlarged to include positive and negative fractions, for the equation 3x=- 2 could not be solved until was understood. In arithmetic the processes

b- a and

b

α

3

2

are limited to positive numbers, but in algebra each of these has a meaning for all values of b and a, positive or negative. This process of generalizing was first explained by the English algebraist, George Peacock (1830), and called by him the principle of permanence. That is, in order that any number or symbol may be made a part of algebra, it must conform to the basic laws governing the processes, namely, The Associative Law, The Commutative Law, and The Distributive Law.

The famous scientist, Sir William Rowan Hamilton (1840), regarded these laws as distinguishing algebraic number from other numbers. In doing so, he discovered numbers which do not obey the Commutative Law of Multiplication, and to these numbers he gave the name Quaternions. Their study has since become a new branch of mathematics.

Hamilton was of Scotch parentage, but Ireland shares his fame, because he was born and educated at Dublin. Like Tartaglia, he received instruction at home when a boy, and showed exceptional ability at an early age. When only thirteen he could read a dozen languages,

at eighteen he had mastered Newton's Principia, and shortly became professor of Astronomy in Trinity College, Dublin. Hamilton did much for mechanics and astronomy, but his greatest achievement in mathematics was the discovery of Quaternions.

CHAPTER II

EQUATIONS

EQUATIONS WITH ONE UNKNOWN

69. Two algebraic expressions are equal when they represent the same number.

70. If two numbers are equal, the numbers are equal which result from:

1. Adding the same number to each.

2. Multiplying each by the same number.

Subtraction and division are here included as varieties of addition and multiplication.

71. The equality of two expressions is indicated by the symbol,=, called "the sign of equality."

72. Two equal expressions connected by the sign of equality form an equation.

73. Such values of the letters as make two expressions equal are said to satisfy the equation between these expressions.

74. Equations that are satisfied by any set of values whatsoever for the letters involved are called identities.

75. Equations that are satisfied by particular values only are called conditional equations, or, when there is no danger of confusion, simply equations.

76. The numbers that satisfy an equation are called the roots of the equation.

77. To solve an equation is to find its roots.

78. The letters whose values are regarded as unknown are called the unknowns.