19. Multiply a2- Jab+b2 by a2+1ab-3b2, and verify the result when a = 1, b = 2. 1 20. Multiply x-xy+y by x1-y1. 21. Divide x+x3y3+y by x* 1 22. Find the product of 2y+y and a-y3. 23. Multiply a-x1 by a3+x3. 24. Divide c-3-Sc-1-3 by c-1-3. 25. Divide х 4x3 y ̃2 – 12x31y-1 + 25 −24x ̃ ̄3y+16x ̃3y2 by 2x3y-1−3+4x ̄3y. 26. Find the value of (ax-2+a ̄1x) (ax ̄2 - 3a ̄1x). 27. Find the square of a1-1-a ̄3 ̧ α 28. Find the continued product of 3a-2-1x, ax3- 1a, and ax3 +b. 29. Divide 2-y by x¦y1+x1y1+x1y1. 30. Multiply a2+2a-2-7 by 5+a2-2a-2. 31. Find the value of (3xy-a − x ̄aya)(xay − x ̄ay ̄1). Important Cases in Division. 66. The following example in division is worthy of notice. Example. Divide a3+b+c3-3abc by a+b+c. a + b + c ) a3 − 3abc+ b3+c3 ( a2 — ab − ac + b2 − bc +c2 Here the work is arranged in descending powers of a, and the other letters are taken alphabetically; thus in the first, remainder a b precedes a'c, and ac precedes 3abc. A similar arrangement will be observed throughout the work. 67. The following examples in division may be easily verified; they are of great importance and should be carefully noticed. and so on; the divisor being x-y, the terms in the quotient all positive, and the index in the dividend either odd or even. II. ́x2 + Y3 = x2 − xy + y2, | x2+y3 = x* − x2y+x2y2 — xy3+y1, =x-x13y+x1y2-x3y3+x2y—xy3+y", and so on; the divisor being x+y, the terms in the quotient alternately positive and negative, and the index in the dividend always odd. III. x+y and so on; the divisor being x+y, the terms in the quotient alternately positive and negative, and the index in the dividend always even. IV. The expressions 2+y2, x2+y1, x2+y3,... (where the index is even, and the terms both positive) are never exactly divisible by x+y or by x-y. All these different cases may be more concisely stated as follows: (1) "y" is divisible by x − -y if n be any whole number. (2) x+y" is divisible by x+y if n be any odd whole number. (3) "y" is divisible by a+y if n be any even whole number. "y" is never divisible by x+y or by x-y, when n is an even whole number. Dimension and Degree. 68. Each of the letters composing a term is called a dimension of the term, and the number of letters involved is called the degree of the term. Thus the product abc is said to be of three dimensions, or of the third degree; and ax1 is said to be of five dimensions, or of the fifth degree. A numerical coefficient is not counted. Thus 8a2b5 and a2b5 are each of seven dimensions. 69. The degree of an expression is the degree of the term of highest dimensions contained in it; thus a-8a3+3a-5 is an expression of the fourth degree, and a2x-7b2x33 is an expression of the fifth degree. But it is sometimes useful to speak of the dimensions of an expression with regard to some one of the letters it involves. For instance the expression ax3-bx2+cx-d is said to be of three dimensions in x. 70. A compound expression is said to be homogeneous when all its terms are of the same degree. Thus 8a6-a4b2+9ab5 is a homogeneous expression of the sixth degree. It is useful to notice that the product of two homogeneous expressions is also homogeneous. Thus by Art. 47, (2a2-3ab+4b2) ( − 5a2+3ab+4b2) = − 10a1+21a3b-21a2b2+16ba. Here the product of two homogeneous expressions each of two dimensions is a homogeneous expression of four dimensions. Also the quotient of one homogeneous expression by another homogeneous expression is itself homogeneous. For instance in the example of Art. 66 it may be noticed that the divisor is homogeneous of one dimension, the dividend is homogeneous of three dimensions, and the quotient is homogeneous of two dimensions. EXAMPLES VIII. c. 1. Divide a3+30ab – 125b3+8 by a-5b+2. 2. Divide 2+ y3 − z3+3xyz by x+y-z. 3. Divide a3- b3+1+3ab by a−b+1. Without actual division write down the quotients in the following cases : 17. In the expression 2a3b2+3ab1+3a2b2x − x5 +20a2b3 - 11a1+7a3b2, which terms are like, and which are homogeneous? 18. In each term of the expression 7a3bc2 - ab2c+12b3c1 — b3c, introduce some power of a which will make the whole expression homogeneous of the eighth degree. 19. By considering the dimensions of the product, correct the following statement (3x2 – 5x1 + y2) (8x2 - 2xy - 3y2) = 24x1 -- 46x2y +9x2y2+13xy3 − 313, it being known that there is no mistake in the coefficients. 20. Write down the square of 3a2-2ab-b2, having given that the coefficients of the terms taken in descending powers of a are 9, 12, 2, 4, 1. 21. Write down the value of the product of 3a2b+5a3 - ab2 and ab2+5a3 - 3a2b, having given that the coefficients of the terms when arranged in ascending powers of b are 25, 0, -9, 6, − 1. 22. The quotient of x3-y3-1-3xy by x-y-1 is x2+xy+x+y2 −y + 1. Introduce the letter z into dividend, divisor, and quotient so as to make them respectively homogeneous expressions of three, one, and two dimensions. CHAPTER IX. SIMPLE EQUATIONS. 71. AN equation asserts that two expressions are equal, but we do not usually employ the word equation in so wide a sense. Thus the statement x+3+x+4=2x+7, which is always true whatever value x may have, is called an identical equation, or briefly an identity. The parts of an equation to the right and left of the sign of equality are called members or sides of the equation, and are distinguished as the right side and left side. 72. Certain equations are only true for particular values of the symbols employed. Thus 3-6 is only true when_x=2, and is called an equation of condition, or more usually an equation. Consequently an identity is an equation which is always true whatever be the values of the symbols involved; whereas an equation (in the ordinary use of the word) is only true for particular values of the symbols. In the above example 3x=6, the value 2 is said to satisfy the equation. The object of the present chapter is to explain how to treat an equation of the simplest kind in order to discover the value which satisfies it. 73. The letter whose value it is required to find is called the unknown quantity. The process of finding its value is called solving the equation, The value so found is called the root or the solution of the equation. 74. An equation which involves the unknown quantity in the first degree is called a simple equation. It is usual to denote the unknown quantity by the letter x. The process of solving a simple equation depends only upon the following axioms: 1. 2. If to equals we add equals the sums are equal. If from equals we take equals the remainders are equal. 3. If equals are multiplied by equals the products are equal. 4. If equals are divided by equals the quotients are equal. |