[a { 7a (8a 9. Simplify: 4a (6a2a-9)}]. 10. Collect coefficients of like terms in: 11. If a = 4x2+7xy+6y2, b = 3xy 8y2 + x2, 2y2 x2 12xy, prove that a+b+c+d 13. Collect the coefficients of like terms in: CX CX3 dx + x2 ax3 - bx2 14. Collect 'the coefficients of like terms in: 16. If A 6z7x+2y, B = 8y3z + 3x, C = 3z2x+5y, D = 4x + 3y - 5z, find the value of CHAPTER V MULTIPLICATION AND DIVISION 36. In Arithmetic, muliplication was defined as the process of taking one number, called the multiplicand, as many times as there are units in the other number, called the multiplier, and the result was called the product. Multiplication may then be considered an addition of a specified number of equal numbers. Thus 3 x7 = 7+7 +7, that is, the number 7 is taken three times. It is evident that multiplication is only a shortened addition of equal numbers. Our definition of multiplication fails when the multiplier is a fraction, for then we divide the multiplicand into as many equal parts as there are units in the denominator and take as many of these parts as there are units in the numerator. has the same re Here we can see that the multiplier lation to a unit as the product 6 has to the multiplicand. Hence we get a more general definition of multiplication, as the process of obtaining the product from the multiplicand in the same way as the multiplier is obtained from unity. Algebraic multiplication differs from Arithmetic because of the use of negative as well as positive numbers. LAW OF SIGNS IN MULTIPLICATION 37. According to our definition of multiplication, since +4 = +1 +1 + 1 + 1, adding 4 units, +6 +6 +6 +6 = 24, adding four plus ·−6 + (-6) + (−6) + (−6) +4 X (-6) Also since 4= −1 − 1 − 1 1, subtracting four units, 4 X (+6) ting four sixes and = −6 6 6 6 = 24, subtrac 4 × (-6) = − (−6) − (−6) − (−6) X -(-6) subtracting four minus sixes equals 6 +6 +6 + = 24. 6 From these multiplications it is evident that if a and b stand for any two numbers, That is when the multiplier and multiplicand have like signs, the product, is plus, but when the multiplier and multiplicand have unlike signs, the product is minus. Therefore, the Law of Signs in Multiplication. Like signs give plus, and unlike signs give minus. THE INDEX LAW IN MULTIPLICATION 38. The exponent of a letter indicates the number of equal factors of the product, thus x3 = x.xx. If therefore, x33 is multiplied by x2, two additional equal factors are put into the product, making five in all, that is, x3 X x2 = X. X. X. X. X = x5. It is evident, therefore, that when two or more factors containing the same letter are multiplied, the exponent of that letter, in the product, will be equal to the sum of the exponents of that letter in the various factors. Thus x3. x2. x4 X3+2+4 = x9. = This is true whether the exponents are numerical or literal, that is xa.x = x+ь. Hence, The Index Law in Multiplication. The exponent of a letter in the product is equal to the sum of the exponents of that letter in the factors. MULTIPLICATION OF MONOMIALS 39. Since the factors of a product may be written in any order, 1. 3x2y X 4xy3 = 3 × 4 X x2 X x X y X y3 = 12x3y1. Note that we only add the exponents of like letters. 2.-5a2b 2b2c-5 X 2 X a2 × b × b2 x c = 10a2 b3c. 3. 3a2 X-2an X-4a2n-4 = 3 × (−2) × (−4) × a2 X an Xa2n-4 = 24a3n-2. Note that the product of more than two factors, called the continued product, is found in the same way as the product of two factors; also that, since every pair of negative factors produce a positive product, our product will be plus when we have an even number of negative factors and minus, when the number of negative factors is odd. Rule for finding the product of any number of monomials. Determine the sign of the product according to the Law of Signs in Multiplication; find the product of the numerical coefficients and to this product annex the letters, giving each letter the sum of the exponents of that letter in the various factors. 40. Degree of a Term.-The degree of a term is determined by the number of literal factors it contains. Thus, 2a is of the first degree, 5ab is of the second degree, and 3ax2y3 is of the sixth degree. Degree of a Compound Expression. The degree of a compound expression is the degree of the term having the largest number of literal factors. Thus, x2 - 2x3y + 4x4y2, is an expression of the sixth degree. Homogeneous Expression.-When all the terms of a compound expression are of the same degree, the expression is said to be homogeneous. Thus, x 2x2y2+y4 is homogeneous because all the terms are of the fourth degree. 41. Dominant Letter.-When one letter in an expression is more important than any other, it is called the dominant letter. Thus, in, x2 + px + q, x is the dominant letter and the expression is said to be of the second degree in x, as the highest power of the dominant letter is the second. Arrangement of a Compound Expression. An expression is arranged according to the powers of some letter, usually the dominant letter, when it is written so that the powers of that letter either ascend or descend. Thus, ax3 2a2x2+2a3xa, is said to be arranged according to the descending powers of x; the highest power of x being written first, then the next highest and so on. But 1 + x + x2 + x3 is arranged according to the ascending powers of x, as the lowest power of x is written first. |