20. The cube of a trinomial can be obtained by the above method, provided that two of its terms be enclosed in a parenthesis and regarded as a single term. (a2 — 2a — 1)3 = [(a2 — 2a) — 1]3 = '(a3 — 2a) 3 — 3( a2 — 2a)2 + 3( a2 — 2a) = a® — 6a3 + 12a-8a3 - 3(a' — 4a3 ANY POWER OF A BINOMIAL. 21. By actual multiplication, we obtain : (a + b)2 = a2 + 2ab+b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b) * = a + 4a3b + 6a2b2 + 4ab3 + b'; etc. (a - b)' = a* — 4a3b + 6a2b2 — 4ab3 + b'; etc. In these results we discern the following laws: I. The number of terms exceeds the exponent of the binomial by one. II. The exponent of a in the first term is identical with the exponent of the binomial; and thereafter decreases by one in each succeeding term. III. b first appears in the second term, having 1 as its exponent which thereafter increases by 1 in each succeeding term. IV. The coefficient of the first term is 1; and that of the second is the exponent of the binomial. V. The coefficient of any term may be found by dividing by the number of preceding terms the product of the coefficient and the exponent of a in the term next preceding. (NOTE. The number of preceding terms is always equal to the exponent of b increased by one.) VI. If the principal sign of the binomial is minus, then the signs are alternately plus and minus, beginning with plus; [e. g. (a — b)' = a3 — 3ab + 3ab' — b3] ILLUSTRATIVE EXAMPLES. 1. Expand (a + b) 3 = a + 5a'b + 10ab + 10a b' + 5ab' + 65 Ans. In the above, as we remarked in Law I, the first term a has 5 for exponent, and the latter decreases by one in each succeeding term. Likewise, b begins with one for exponent, which increases to five at the last. With regard to coefficients, we have 5, the exponent of the binomial for coefficient of the second term; and to obtain the coefficient of the third, we multiply that of the second by the exponent of a in that term, and divide that result by the number of terms preceding the one sought. 2. Expand (3m — - n2)* = [(3m) - (n2)]* = (3m) — 4(3m)3 (n2) + 6(3m)2 (n3)2 — 4(3m) (n2)3 + (n2)' =81m-108m3n2 + 54m2n' — 12mn® + no. Ans. EVOLUTION. 22. A root of a quantity is one of the equal factors into which it is resolved. Evolution is the process of finding any required root of a quantity. It is accomplished by finding a quantity which, on being raised to the proposed power, will produce the given quantity. The radical sign is . When it is prefixed to a quantity it indicates that some root of that quantity is to be found. ✔a indicates the second or square root of a. 3 a indicates the third or cube root of a. Va indicates the fourth root of a. 'a indicates the fifth root of a. And so on. The figure written over the radical sign is called the index. If no index is written the square root is understood. EVOLUTION OF MONOMIALS. 23. Find the cube root of ab3co. We must find a number which, raised to the third power, will produce a b'c. The required number is evidently a2bc3. Therefore a°b3c9 = a2bc3 To give the above principle in words: Any root of a monomial is obtained by dividing the exponent of each factor by the index of the required root. We have observed that odd powers have the same sign as the quantity itself, while even powers have the positive sign. Hence, 1. The odd roots of a quantity have the same sign as the quantity itself. 2. The even roots of a positive quantity are written positive or negative. 3. The even roots of a negative quantity are impossible. This is evident because no quantity raised to an even power can produce a negative result. Such roots are termed imaginary quantities. We may now state the following rule for determining the roots of a monomial. Extract the required root of the numerical coefficient, and divide the exponent of each letter by the index of the root. Give to any even root of a positive quantity the sign +, and to any odd root of a quantity the sign of the quantity itself. Any root of a fraction may be determined by taking the required root of each of its terms. Find the square root of 16aRb3m √16a®b3m = +4a3bm 10 32x10 243y" Find the fifth root of Find the square root of 9x1y2z® By the rule, 9x*y*z* = +3x2yz' Ans. 10 NOTE. To obtain the root of any fraction, extract the required root of each of its terms. SQUARE ROOT OF POLYNOMIALS. 24. Since (x + y)2 = x2 + 2xy + y2, we know that x + y must be the square root of a2 + 2xy + y2. What process will determine this to be true, when the latter alone is given? x2 + 2xy + y2 2x+y2xy + y2 2xy + y2 x + y The square root of the first term is x, which is the first term of the root. Subtracting its square from the given expression, the remainder is 2xy + y2, or (2x + y) y. Dividing the first term of the remainder by 2x, or twice the first term of the root, we obtain y, the second term. Add this to 2x to obtain the complete divisor 2x + y; multiply this by y, and the product 2xy + y2, subtracted from the remainder, completes the operation. From the above process we derive the following rule: The terms should be arranged according to the powers of some letter. The square root of the first term will be the first term of the root; after finding it, subtract its square from the given expression. Divide the first term of the remainder by twice the first term of the root, and add the quotient to the root and also to the divisor. Multiply the complete divisor by the term of the root last obtained, and subtract the product from the remainder. If other terms remain, proceed as before, doubling the part of the root already found, for the next trial divisor. Arranging according to the descending powers of x, 9x + 12x3 — 8x1 — 14x3 + 4x + 1 | 3x3 + 2x2 — 2x — 1 92 - Ans. Notice that each trial divisor is equal to the preceding complete divisor, with its last term doubled. 2. 40a25-14a2+9a* - 24a3 THEORY OF EXPONENTS. 25. We have heretofore considered only positive integral exponents. It is often convenient to employ fractional and negative exponents. We defined the positive integral exponent as indicating the number of times a quantity was taken as a factor, thus, m and n representing any positive integral numbers, for example 4 and 3 giving a* × a3 = a* +3 = a”. |