The Highest Common Factor by the Method of Division Solutions of Quadratic Equations ELEMENTARY ALGEBRA SECOND YEAR COURSE CHAPTER I ELEMENTARY DEFINITIONS AND OPERATIONS FUNDAMENTAL DEFINITIONS 1. In arithmetic, numbers are commonly represented by Hindu-Arabic numerals. In algebra, numbers are represented also by letters. Any combination of numerals, letters and symbols of operation, which stands for a number, is called an algebraic expression. An algebraic expression may consist of parts which are separated by the or signs; these parts, with the signs immediately preceding them, are called terms. Thus, in + a2 -— 2 ab + 3 b2, there are three terms, + a2, — 2 ab, +3 b2. Each of the numbers which multiplied together form a product is called a factor of the product. A factor consisting of one or more letters is called a literal factor. Terms which have the same literal factors are called similar; terms which do not have the same literal factors are called dissimilar. 3 ab2 - 10 a2b, the terms + 4 a2b and 10 a2b are similar; 3 ab2 and - 10 a2b are dissimilar. An algebraic expression of one term is called a monomial, of two terms a binomial, of three terms a trinomial, and of several terms a polynomial. When a number is the product of two factors, either factor may be called the coefficient of the other factor. Often the word coefficient is applied only to the factor which is expressed in numerals. Thus, in 5 xyz2, the numeral 5 is called the coefficient of xyz2. An exponent is a number placed at the right and a little above another number, called the base. When the exponent is a positive integer, it indicates how many times the base is taken as a factor. The exponent expresses the power to which the number is raised. Thus, the 4 in a expresses the fourth power of a. Find at when a = 2. Has at the same value as 4 a, when a = 2? The absolute value of a number is its value regardless of its sign. Thus +5 and 5 have the same absolute value, 5. ADDITION AND SUBTRACTION 2. Addition: If similar terms have like signs, find the sum of the absolute values and prefix the common sign. If similar terms have unlike signs, find the difference of the absolute values and prefix the sign of the one which has the greater absolute value. If the terms are dissimilar, the addition is indicated in the usual way. Thus, the sum of 3 a and 5 b is 3 a + 5 b. To check an example in addition: Substitute numerals for the letters and find the sum. This sum must equal the result of substituting the numerals in the answer. 3. Subtraction: Conceive the sign of the subtrahend to be changed, and then proceed as in addition. Check by substituting numerals for the letters. In what other way may subtraction be checked? WRITTEN EXERCISES 4. Find the sum of: 1. 7a-3b+6c; 4a+5b-12 c; 6a+2b+3c. 2. 4x2-3x2y - 4xy2; 3x2y - 8xy2-8x2; 3x2 - 6 x2y. 3. 5a" - 6 ab" — 7 c"; 2 a" +3 a′′b" +8c"; 4. 3(a - b)+4(a + b); 7(a − b) — 5(a + b) ; - 5 a"b" - 2c". — 6(a — b) 5. a4-3 a3 +9 a2-4a+6; 7 a1 + 3 a3 +2 a2 + 10 a +7; 7. From 2 a2 2 ab+3b2 take a2 ab - b2. 8. From 4+3c-8d-9e take 7c+5e-10-2 d. 9. From a + 2 a*b1 6 by take 2 a* 5 ab2 + 8bo. 10. From 3(x + y) −7(x − y) take 4(x − y) − 5(x + y). 11. From (xy)a — (x + y)b take (x + y)a +(x − y)b. 12. What number added to 7x2y-5xy2-6xy will give 2 xy2 — 3x2y + 4 xy? ORDER OF FUNDAMENTAL OPERATIONS 5. The values of many algebraic expressions depend upon the order in which the operations of addition, subtraction, multiplication, and division are performed. To avoid confusion it has been found necessary to adopt certain rules, so that an expression shall always be interpreted in the same way and |