EXPONENTS IN MULTIPLICATION 59. An exponent is a symbol, numerical or literal, written above and to the right of a given quantity, to indicate how many times that quantity occurs as a factor. Thus, if three a's occur as factors of a number, we write a3, and avoid the otherwise cumbersome form of a x axa. In like manner, axaxax b× b = a3b2, and is read "a cube, b square." 60. The product of two or more equal factors is a power. Any one of the equal factors of a power is a root. In common practice, literal and other factors having exponents greater than 3 are read as powers. Thus, a is read "a sixth power," or merely The exponent "1" is neither written nor read. That is, a is the same as a1. The difference between coefficients and exponents must be clearly understood. A numerical illustration emphasizes that difference. Thus: Similarly, a3 = a xa xa, at = ахахаха. α8 x α = αχαχαχαχαχαχα = a1. a5 × a1 = aб+4 = ao, m2 x m3 x m = m2+3+1 = m6. In general, therefore, we have the following: If m and n are any positive integers : ama x ax a... to m factors, a" = a xa xa... to n factors. Hence, am × an = (αχαχα to m factors) (αχαχα In the same manner, am × an × a2 = am+n+p, and so on, indefinitely. This principle establishes the first index law, m and n being positive and integral. The general statement of this important law follows: 61. The product of two or more powers of a given factor is a power whose exponent is the sum of the given exponents of that factor. Oral Drill (See also page 388) MULTIPLICATION OF A MONOMIAL BY A MONOMIAL By application of the law of order and the principles for signs and exponents, we obtain a process for the multiplication of a monomial by a monomial. Illustrations: 1. Multiply 3 a2b3 by 12 a12. By the Law of Order, 3 a2b3 y. 1.2 a4b2 = 3 x 12 x a2 × aa × b3 × b2. By the Law of Grouping, By Arts. 58 and 61, = =(3 × 12)(a2 × aa) (b3 × b2). = 36 a6b5. Result. == 2. Multiply -7 a2b3×3 og 5 a7b2y. -'7 a2b3x3z × 5 a7b2y = −7 × 5 × a2 × a7 × b5 × b2 × x3 × 2 × y − (−7 x. 5) (a2 × a3) (b5 × b2) (x3)(z) (y) Therefore, to multiply a monomial by a monomial: (53) (54) (57) (61) 62. Observing the law of signs, obtain the product of the numerical coefficients. The exponent of each literal factor in the product is the sum of the exponents of that factor in the multiplicand and multiplier. MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL The process of multiplying a polynomial by a monomial results directly from the number principle for multiplication assumed in Art. 55. That is: a(x + y + z) = ax + ay + az. In common practice the multiplicand and multiplier are written as in arithmetic, excepting that the multiplier is usually written at the extreme left. Illustration: Multiply 3 m3— 5 m2n +7 mn2 − 2 n3 by — 2 mn. 3 m3 - 5 m2n + 7 mn2 - 2 n3 -2 mn −6 man + 10 m3n2 – 14 m2n3 + 4 mn Result. Each term of the prod uct is obtained by the principles of Art. 62, for the operation is made up of successive multiplications of a monomial by a monomial. Hence, to multiply a polynomial by a monomial: 63. Multiply separately each term of the multiplicand by the multiplier, and connect the terms of the resulting polynomial by the proper signs. 7 a3x2 is a term of the 5th degree, for 3 + 2 = 5. 65. The degree of an algebraic expression is determined by the term of highest degree in that expression. 5 m2n + mn + n2 is an expression of the 3d degree. 66. An algebraic expression is arranged in order when its terms are written in accordance with the powers of some letter in the expression. If the powers of the selected letter increase from left to right, the expression is arranged in ascending order. Thus, х -2x2+5 x3- 7 x2+10x5. If the powers of the selected letter decrease from left to right, the expression is arranged in descending order. Thus, 4x95x7 + 3 x5 – 2 x3 – 3 x. 67. The degree of a product is equal to the sum of the degrees of its factors. 68. A polynomial is called homogeneous when its terms are all of the same degree. Thus, x2 - 4x3y + 6 x2y2 − 4 xy3 + y1 is a homogeneous polynomial. MULTIPLICATION OF A POLYNOMIAL BY A POLYNOMIAL A further application of the law of distribution for multiplication (Art. 55) establishes the principle for multiplying a polynomial by a polynomial. By Art. 55: (a + b) (x + y) = a (x + y) + b (x + y) = ax + ay + bx + by. The polynomial multiplicand, (x+y), is multiplied by each separate term of the polynomial, (a+b), and the resulting products are added. The process will be clearly understood from the following comparison: 156 100+ 50+ 6 Explanation: 10 (10 + 2) = 100+ 20 (10+2) = 30+ 6 100+ (20+30) + 6 = 156 a + 5 a + 7 a2 + 5 a + 7 a +35 Multiplicand. a2 + 12 a + 35 Product. Explanation: a (a + 5) = a2 + 5 a 7(a + 5) = 7 a +35 a2 + (5 a + 7 a) + 35 = a2 + 12 a 35 We have, therefore, the following general process for multi plying a polynomial by a polynomial : 69. Arrange the terms of each polynomial according to the ascending order or the descending order of the same letter. Multiply all the terms of the multiplicand by each term of the multiplier. Add the partial products thus formed. |