duct of unlike signs gave minus and the product of like signs gave plus; e. g. (- 5X)X(— 3Y)X(-2Z). Here the product of like signs gives minus. Lesson XI.-Removal of Parentheses and Multiplication of Monomials by Polynomials. Examine the expression 5B(AX+3A). What sign is understood between the B and the parenthesis? (The multiplication sign) Hence, to remove the parenthesis, every term within it must be multiplied by the 5B. What is the value of the above expression when the parenthesis is removed? Again, take the expression 3A(4BX-2AB). Here also each term enclosed must be multiplied by the 3A, the result being 12ABX-6A3B. Suppose we have --5A(3AB+2ST—A3Y). By what must each enclosed term be multiplied? Answer, by --5A. Perform the multiplication. In the expression X2, what is the coefficient? Answer, I understood. I understood. In the quantity - (7AB-3AX- 5 BC), what is the coefficient of the parenthesis? Answer, 1. What is its sign? Answer, Minus sign. To eliminate the parenthesis we must multiply each term within by the minus one (-1). What will then be the value of the parenthesis? Exercise. Simplify the following expression by removing the parentheses and combining similar terms: 5(AX+BY) — 3A(X—B) + 2B(A—Y), etc. 3(AX-BY) — (2AX—4BY) + À (X−−Y) —(AY+2X), etc. − 4A (— 3AB+2A3B2 — 5AX). Suppose you are asked to multiply 5ABX -- 2 A3 -- 12AX2 -- 4B5 by A2BX3, how could you indicate the multiplication? Answer, By placing the polynomial in parenthesis and the monomial in front. How could you find the result? By eliminating the parenthesis. But what must you do to eliminate the parenthesis? Multiply each term in the parenthesis by the monomial. We thus arrive at the follo wing law: The product of a polynomial by a monomial is an expression which is composed of the sum of the partial products obtained by multiplying each term of the polynomial by the monomial. Show inductively the following scholia: Scholium I-There will be as many terms in the product as there are in the multiplicand. Scholium II.--The signs in the product will all be the same as in the multiplicand, if the multiplier is positive, and all will be reversed if the multiplier is negative. Scholium III.-The degree * of each term in the product will be * Explain that the degree of the term is indicated by the number of variable factors contained in it. Thus 8A'B' is of the fifth degree because it contains as factors 8AABBB or five variable factors. Let the pupil discover the method of finding the degree of a term by adding exponents (including, of course, the "1" understood, where t occurs). equal to that of the multiplicand increased by as many points as there are in the degree of the multiplier. The above scholia should be used to test each example after it is completed, in the following manner: First, the number of terms in the product should be counted and found to agree with the number in the multiplicand. Second, the signs of each term in the product should be examined and found either to agree or disagree entirely with those of the multiplicand. Third, the degree of each term should be examined and found to differ from the degree of the corresponding term in the multiplicand by the degree of the multiplier. The form of work may now be altered to conform with the appearance of an example in multiplication in Arithmetic; that is, the multiplicand may be placed above the multiplier and a line drawn. under which the product is placed. Lesson XIII.-Multiplication of Polynomials by Polynomials. What is the result of removing the parenthesis from the following expression? Q (A-2AB2+4AB3). Answer, AQ-2 AB Q+4A B3Q. Suppose, instead of the general number Q, we used any other general number, what would be the effect? Answer, "That general number would appear where the Q had before appeared. Is (AX) a general number? (Cf. Lesson on Parenthesis.) Suppose that for Q we substitute (A-X) and obtain the result (A-X) (A3 - 2 AB2+4A B3), could the second parenthesis now be removed? How? Answer. A3 (A—X)—2AB (A—X)+4AB3 (A—X). (Exercise in removing the second parenthesis from products in which the first parenthesis is a binomial and the second parenthesis is a trinomial.) Could all the parentheses be removed from the above expressions? How? Exercise.--Remove all the parentheses from the expressions obtained in the last exercise. When we examine the answers to the above, how many terms are there in the product? (6.) How many terms in the first parenthesis? (2.) In the other parenthesis? (3.) How were the six terms in the product obtained? If this question is not answered correctly at first, and it is not likely to be so answered, repeat the operations involved and analyze the product, term for term, until finally the pupil sees that a product of a binomial by a trinomial is obtained by multiplying each term of the multiplicand by each term in the multiplier. Now take exercises involving trinomial multipliers and polynomial multiplicands and then deduce the following law: The product of a polynomial by a polynomial is an algebraic expression composed of the sum of all the partial products obtained by multiplying each term of the multiplicand by each term of the multiplier. Let us now arrange an example to conform in appearance with the ordinary arithmetical example in multiplication; e. g., X+Y+Z By what mechanical means may we be sure that each term of the multiplicand is multiplied by each term of the multiplier? Answer, By multiplying the entire multiplicand, first by A and then by B. Suppose we had three terms in the multiplier and four in the multiplicand, how many terms would there be in the product? Answer, Twelve. Multiply by R+S+T+U AR+AS+AT+AU+BR+BS+, etc. Suppose some of the terms in the answer were similar, what could we do to simplify? Example: multiply (R+S+T+C) by (B+C+R). What terms in the answer are similar? (CR.) Could the answer, then, be simplified? How? Example: Multiply A2+AB+B2 by A+B How can this answer be simplified? Could anyone suggest a means of simplifying the entire operation? How? Answer, By placing similar terms under each other immediately and adding. Example: Multiply 3A2+AB+B2 by A2+2AB+3 B2 -3 3A +2A B+A2B2 6A3B+4A3B2+2AB3 9A2+B26AB3+3 B+ 3A18+A3B+14A2B2+8AB3+3B Answer. Example: Multiply 2AB+3A2+B2 3B2+A2+2 AB 6AB3+9A2B2+3 B+ A2B2 +2AB+3A' 2 AB3+4A2B2 +6A3B 8AB3+14A2B2+3B1+8A3B+3A' Answer. What is the difference between the two foregoing examples? Which arrangement produces the simplest result? Hence the following rules: First. Arrange both multiplicand and multiplier according to the descending power of the dominant letter (i. e., the one that appears most frequently). Note: If two letters appear with equal frequency, prefer that one which comes first in alphabetical order. Second.--Find the partial products of the multiplicand by each term of the multiplier according to the rules for multiplying polynomials by nomials. Third. Place these partial products under each other, similar terms in columns where possible. Fourth. Add these partial products together and their sum will be the required answer. Scholium.-Test each partial product according to the scholia under Multiplication of Polynomials by Nomials. Exercise with many examples, first giving them with all the terms properly arranged, later with the terms in confused order. The remaining seven lessons will take up Division, Problems, and Factoring. The New York Society for the Scientific Study of Education SIDNEY MARSDEN FUERST, I. EDWIN GOLDWASSER, JAMES A. O'DONNELL, President Examination for Principal's License AND Assistant to Principal's License Beginning in March, 1905, under the auspices of the abovenamed society, courses of lectures will be given to teachers who desire to enter examinations for higher licenses, viz: Assistant to Principal's License, and Principal's License. Courses of Lectures for Men (Candidates who intend to enter the next examination for license as Principal of Elementary Schools) will include the following subjects: History and Principles of Education Methods of Teaching School Management Rhetoric and English Literature These courses will begin in March. Classes will be limited in number. Applications should be filed at once. SIDNEY MARSDEN FUERST 21 East 14th Street, New York No teacher will dispute the value of GOOD MUSIC as an aid to study and discipline. Singing and marching form a part of the day's program in every modern school-room. No properly conducted school is considered completely equipped if it lacks a musical instrument. The REGINA is a high-grade instrument of American manufacture, which makes an acceptable substitute for the piano where the latter is absent or where there is no one to play. The tone of the REGINA is rich and sweet and loud enough for a school-room of any size. It plays patriotic, sacred or classical music with equal facility. There are thousands of tunes to select from. Its The REGINA keeps time perfectly. music is successfully used for singing and marching, also for furnishing that rhythm which is so necessary while pupils are performing calisthenic exercises or practicing penmanship. There can be no doubt that the REGINA is a valuable aid in teaching. Its success has been demonstrated. Office of JAMES OTIS KALER, Supt. of Schools, SOUTH PORTLAND, ME, March 17, 1904. The Regina Company, New York. GENTLEMEN:-We have had our REGINA about four weeks and each school begs for the loan of it. The continuous tune discs "work like a charm." The waltz music is good for dumbbell exercises, and we are using "El Capitan" March for club-swinging. I have found the instrument especially valuable in calisthenic exercises where perfect time is of importance. I have also used the REGINA for the children to sing by, and am well satisfied that it is an aid to the music teacher. Late in the afternoon, when the children get restless, some song is played two or three times and the work goes on the better for it. One of our schools gives a concert next week with the REGINA, and the pupils are selling tickets of admission at 25 cents each. I am enthusiastic in regard to the use of the REGINA in schools, and have difficulty in deciding whether it is any more valuable in one grade than in another. From the Ist to the 9th, it is helpful, to say nothing of the pleasure the children may have from it during visitation days, at public examinations, and in the regular school entertainments. I am thoroughly satisfied that a music box like the REGINA is better for ordinary school-room purposes than a piano, and I have given it a good trial. Besides, the cost saved in tuning a piano would provide each REGINA with at least one dozen discs every year. I believe there is not a school in the country that wouldn't use a REGINA as I am doing if the advantages could be properly presented. Only teachers themselves can appreciate the assistance which such an instrument would be to them. Yours respectfully, JAMES OTIS KALER, The above cut shows Regina Hall Clock, No. 2, which may, if desired, be adjusted to play every hour or every half hour. Reginas are made in many styles at prices ranging from $10 to $400. Catalogues, prices and full information furnished on application. THE REGINA COMPANY Main Office and Factory: RAHWAY, N. J. Branches: 11 East 22nd Street, New York; 259 Wabash Avenue, Chicago |