Since a square root is one of the two equal factors and a cube root is one of the three equal factors of a number, any ROOT of a monomial quantity is obtained by extracting the required root of the numerical factors, and dividing the exponents of all the literal factors by the index of the root. in which the exponents of the letters were divided by the root index 2. Similarly, any POWER of a monomial is obtained by raising the numerical factors to the required power, and multiplying all the exponents of the literal factors by the exponent of the power. Thus (3x2y+z-53 27x6y122-15 = 202 826 176. Miscellaneous Examples. Express each of the following in one or more different forms: CHAPTER XII THE BINOMIAL THEOREM 177. A New Symbol. In the study of this chapter we are to derive and apply a formula whose expression is simplified by the use of a symbol not before used. In this formula we shall wish to indicate 1 times 2, 1 times 2 times 3, 1 times 2 times 3 times 4, and so on. Whenever this is necessary in mathematics, that is, when we wish to indicate the product of the integers in succession from unity, instead of writing 1×2×3×4 or 1.2.3.4 and so on, we write 4! or 4, each of which means 1×2×3×4 or 1.2.3.4, and is called factorial 4. Likewise 5! or 5 is called factorial 5 and means 1X2X3X4X5. In modern books the first or exclamation point form is used almost exclusively to denote a factorial number instead of the older or angle form shown in the two preceding illustrations. As the exclamation form is easier to make and is in common use it is preferable to the angle form. Our new symbol, therefore, like other operation symbols, is significant only when written in the proper position with respect to the quantities affected by it. Following a word or sentence it denotes an exclamation; following a symbol for quantity it denotes that the symbol is factorial and therefore integral. In the derivation of 178. An Example in Addition. the formula referred to in the preceding paragraph it will be necessary to add terms involving a factorial denominator, like Observe that with respect to a and b these are like terms because a and b have the same respective exponents. They may therefore be added by adding their coefficients (2) By removing the parenthesis and then adding. In like manner, add and simplify 179. Expansion. When an algebraic expression is operated upon as denoted by an exponent it is said to be expanded and the operation is called expansion. 1. (a+b)2 can be expanded by what law of numbers? Show the application of this law to the expression. Expand the following, showing the full multiplication: 4. (a+b)5. 2. (a+b)3. 3. (a+b)*. Each expansion has how many more terms than the number of units in the exponents of the respective powers? Therefore the expansion of (a+b) 12 would have how many terms? The expansion of (a+b)" would have how many terms. provided n is a positive integer? The first term of the binomials is in all terms of the expansions, except what term? The last term of the binomials is in all terms of the expansions, except what term? Which term of the binomials has increasing exponents in the expansions? Which has decreasing exponents? What is the increase in each succeeding term? What is the decrease? In (a+b)2 what is the first term of the expansion? (a+b)3, (a+b)4, (a+b)5, (a+b)14? How do the exponents of these first terms compare with the respective exponents of the binomials? Therefore what is the first term of the expansion of (a+b)"? How do the coefficients of the second term of the expansions compare with the exponents of the respective powers? The exponent of a is how much less in the second term of the expansions than in the first term? What is the exponent of b in the second term of all the expansions? Therefore what is the second term of the expansion of (a+b)n? 180. Determination of the Coefficients of the Terms of a Binomial Expansion by Inspection. What is the coefficient of the third term of the expansion of (a+b)3? Show by expansion of (a+b)3 whether this coefficient equals the following: Coef preceding termXExp first letter in the term Number of the term Apply the same test to the coefficient of the third term in the expansion of (a+b); also to the coefficient of the third term of (a+b)5. Apply the same test to the coefficients of all the other terms except the first in all of the expansions. LAW FOR THE COEFFICIENT. Write this law by answering the following question: The coefficient of any term except the first in a binomial expansion, equals what? 181. Summary of Facts Regarding the Expansion of a Binomial. In the work-book fill in the following outline: In every binomial expansion. (1) Number of terms = = (3) Exponent first letter in first term = (4) Exponent second letter in first term = (5) In each succeeding term, decrease in exponent first letter = Increase in exponent second letter = (6) Coefficient any term except first = 182. The Binomial Theorem. This celebrated theorem is a law for the expansion of a binomial. If written it would state substantially what has been summarized in paragraph 181, provided your summary is correct. The theorem is true for all exponents whether integral or fractional, positive or negative. It will be best understood as summarized and therefore need not be written in the work-book. |