c (a+b). Factors, expressed by figures, are called numerical factors, and those, expressed by letters, are called literal factors. 6. Coefficients.-Any factor of a number may be considered a multiplier of the other factors and is then called the coefficient or co-factor of this number. Thus, in the number 6ab, 6 may be called the coefficient of ab, or, a may be called the coefficient of 6b, or b may be called the coefficient of 6a. Here again we distinguish between numerical coefficients and literal coefficients. When a literal number has no figure preceding it, the numerical coefficient 1 is understood. Thus, xy has the numerical coefficient 1 understood, since xy is equal to 1 X xy. 7. Powers. When a number is the product of two or more equal factors, it is called a power of the factor. Since 8 is equal to 2 × 2 × 2, it is called a power of 2. Similarly, a X a is called a power of a. To avoid writing all the equal factors of such a product, a small figure, called the exponent, is written to the right and slightly above one of these factors, showing how many equal factors the product contains. Thus, 23 means 2 × 2 × 2 and a2 means a X a. These powers are named according to the number of such equal factors. Thus a2 is called the second power of a; 23 is the third power of 2, b4 is the fourth power of b, x1o is the tenth power of x. The second power of any factor, as b2 is called the square of b, because if b represents the number of units of length in the side of a square, b× b will give the number of square units or units of area in the square. For a similar reason, x is called the cube of x, since, if x represents the number of units of length in the side of a cube, x represents the number of units of volume in that cube. х x Care should be taken to distinguish the meanings of coefficient and exponent. Thus, in 3x, 3 is the coefficient of x and shows that three x's are added to produce 3x, that is 3x = x + x + x, but the exponent 3 of x3 shows that three x's are multiplied together, that is x3 = x. x. x. 8. Roots. When a number is the product of two or more equal factors, each of these factors is called a root of that number. Thus 2 is a root of 8, x is a root of x2. A root of any number is indicated by the root sign or radical sign, ✓, placed over that number, and a small figure, placed in the sign, called the index of the root, shows what root of the number is to be taken. Thus V means the third or cube root of xo; Vy3, means the fifth root of y5. If no index is written, 2 is understood, thus the square root of a is written, Va, the index 2 being understood. 9. Algebraic Expressions.-Any number written with algebraic symbols is called an algebraic expression and may consist of one symbol or two or more symbols connected by signs of operation. Thus, x, 2ab, r + st, are algebraic expressions. Terms. The parts of an algebraic expression, which are separated from each other by either a plus sign or a minus sign, are called the terms of that expression. Thus, b2 in the expression 2ab+ 2ab and " b2 C are the two terms. Note that the parts of a term may be connected by signs of multiplication or division, but not by the signs or . Similar Terms.-The terms of an algebraic expression, which contain the same letters with the same exponents, are called similar terms or like terms. Thus, 2a2b3, a2b3, 7a2b3 are similar terms. Monomial. An algebraic expression consisting of only one term, is called a monomial or simple expression. Thus, xy, ab, or are monomials or simple expressions. 2s Polynomials. An algebraic expression of two or more terms is called a polynomial or ccmpound expression. Thus, ut at2, a2+2ab+b2, are polynomials. An expression of two terms is called a binomial and one of three terms, a trinomial. 10. Numerical Value of an Expression.-If in an algebraic expression, particular values are placed for the letters and the indicated operations are performed, we obtain the numerical value of the expression. 1. If x = 2, find the numerical value of 6x and 3x3. Substituting the value 2 for x, 4. If u = 2, v = 5, find the numerical value of √9uv. As no vinculum or parenthesis or other sign of aggregation is used here, the radical sign belongs only to the symbol immediately following it. Note the difference between this problem and the preceding. Note. If one factor of a product is equal to zero, the product will equal zero, since any number times zero is equal to zero. NUMERICAL VALUES OF POLYNOMIALS 11. In finding the numerical value of a polynomial, find the value of each term, as in the preceding exercise, and then add or subtract the several terms as indicated by the signs or between the terms. In algebra, it is usual + to proceed from left to right, not only in finding the value of each term, but also in combining the several terms. It is evident, that the terms of an algebraic expression may be arranged in any order without altering the value of the expression. This is called the commutative law for addition and subtraction. Thus, 4 + 6 + 3 is equal to 6 + 4 + 3 or to 3+ 6 + 4. = 1. If a 2, b = 3, c = 4, find the value of a2 + 2bc Substituting the values of the letters, 2. If x = 5, y = 4, z= 0, find the value of (x + z)(x + y) + √y. (x + z)(x + y) + √y = (5 + 0)(5 + 4) + √√4 = 5 × 9 + 2 = a + b b+c 5, x = 0, find value of substituting the values of the letters, X √52×2× 32 ÷ (5 − 2 X 1) - 5 X 2 X3 X5X0= EXERCISE 2 2, b = 3, c = 4, d = 5, x = 6, y = 3, z = 0, find 1. When a = find value of REVIEW EXERCISE I 3, c = 10, m = 4, n = 1, k = 0, 2m 5, b = +a2b2abk. 3, p = 1, z= 0, find value of 2 (m + s)2 + 2msp + 3. If x 6, y = 3, z = 0, r = 2, s = 10, find the value of = (x − y)2 + 2(x − y) (x + y) + (r + s)2 + 7 xyz2 4. If a 1, b = 2, c = 3, d = 4, find value of = 5. If a 3, b = 2, x = 12, find value of = 7. If a 5, b = 2, c = 3, d = 4, find the value of = b = 2c α 2, n = 1, x = 4, find the value of 75 - ab2n3 x2 a 12. When a = 5, 3, c = 10, m = 4, find the value of ma-b and 4ac2m. |