same degree. Thus, the terms Sa3bx and y3z are homo geneous, as are the terms a3b2c and 7x3y3. 13. A Polynomial is homogeneous when all of its terms are homogeneous. Thus, the polynomial a3bc 7ax +363c2 is homogeneous, whilst the polynomial 8c3b -7a2c+8x2 is not homogeneous. Two terms are similar or like, when the combination of literal factors is the same in both. Thus, the terms 8x2yz and 7x2yz, are similar, as are also the terms 25a2bcd3 and 2a2bcd3. In order that two terms may be similar, they must contain the same letters, and each letter must have the same exponent in both. The terms 8a2b and 7ab2 contain the same letters, but are not similar. 14. The Reciprocal of a quantity is 1 divided by that quantity: thus, 1 1 a quantities a, a + b, с a+b2 a' ∞, are reciprocals of the duct of any quantity by its reciprocal, is equal to 1. 15. The Numerical Value of an expression, is the result obtained by assigning numerical values to cach letter which enters it, and then performing all the operations indicated. Thus, the numerical value of the expression, ab + ac + bc, when a = 2, b = 3, and c = 4, is 2 × 5 + 2 × 4 + 3 x 4 = by performing the operations indicated 26; EXAMPLES. Find the numerical values of the following expressions, when a = 2, b = 3, c = 4, and d = 5. Find the numerical values of the following expressions, when a 5, b = 2, c = 4, and d = 3. 16. ADDITION is the operation of collecting several quantities, and finding the simplest expression for their aggregate. This expression is called the Sum.. 17. When the quantities are similar, the addition may be performed; when they are not similar, the operation can only be indicated. Thus, the sum of 7a2b and 4a2b, is 11a2b; in the same way that the sum of 7 pounds and 4 pounds, is 11 pounds. The sum of 3a2c and 462, cannot be expressed by a single term, any more than the sum of 3 pounds and 4 feet; the sum is indicated, however, by writing the quantities one after the other, with their proper signs; thus, 3ac2 + 4b2. 18. To deduce a rule for adding similar terms, let us take the following examples: In the first example, the quantity a2bc is taken posi tively, 7+1 + 3 + 5, ΟΙ 16 times, and the sum is +16a2bc. In the second example, the quantity is taken. negatively, 2 + 3 + 1 + 8, or 14 times, and the sum is - 14a2bc. In both cases, the sum is found by adding the coefficients, annexing the common literal part, and prefixing the common sign. In the third example, the quantity a2bc is taken positively, 47, or 11 times, and negatively, 25, or 7 times, and the sum is 11 - 7, or 4 times a2bc. In the fourth example, the quantity a2bc is taken, additively, 53, or 8 times, and negatively, 8+2, or 10 times, and the sum is 8-10, or 2 times a2bc. In like manner, all other groups of similar quantities may be added. Hence, the following rule for adding similar quantities. RULE. Add the coefficients of the positive terms and of the negative terms separately; subtract the less sum from, the greater, prefixing the sign of the greater; then annex to this result the literal part. 19. In adding polynomials together, which contain several groups of similar quantities, it will be found more convenient to write them down, so that each group of similar quantities may fall in a column by itself. We may, therefore, write the following rule for the ad lition of algebraic quantities: RULE. 1. Write down the quantities to be added, so that similar terms shall fall in the same column. |