RULE. Multiply each term of the first expression by each term of the second. When the terms multiplied together have like signs, prefix to the product the sign +, when unlike prefix Example 1. Multiply x+8 by x+7. The product =(x+8) × (x+7) = x2+8x+7x+56 = x2+15x+56. The operation is more conveniently arranged as follows: x + 8 NOTE. We begin on the left and work to the right, placing the second result one place to the right, so that like terms may stand in the same vertical column. Example 2. Multiply 2x-3y by 4x-7y. 41. When the expressions to be multiplied together contain more than two terms, a similar method may be employed. Example. Find the product of 2a2 – 3ab+4b2 and − 5a2+3ab+4b2. 2a2-3ab+462 -5a2+ 3ab+462 -10a4+15a3b-20a2b2 +6a3b-9a2b2+12ab3 42. If multiplier and multiplicand are not arranged according to powers ascending or descending of some common letter, a rearrangement will be found convenient. Example. Multiply 2xz-22 + 2x2 - 3yz+xy by x −y+2z. 43. When the coefficients are fractional we use the ordinary process of Multiplication, combining the fractional coefficients by the rules of Arithmetic. 20. 21. 22. ab+cd+ac+bd and ab+cd-ac-bd. 23. a2-5ab-b2 and a2+5ab+b2. 25. 24. x2-xy+x+y2+y+1 and x+y−1. a2+b2+c2-bc-ca- ab and a+b+c. x3y+y+x2y2+x^ — xy3 and x+y. 26. 27. x12 − x9y2+x®y1 − x3y®+y3 and x3+y2. 28. 3a2+2a+2a3+1+a1 and a2-2a+1. 29. -ax2+3axy2-9ay and -ax-3ay2. 30. -2x3y+y1+3x2y2+x2 − 2xy3 and x2+2xy + y2. 44. Although the result of multiplying together two binomial factors, such as x+8 and x-7, can always be obtained by the methods already explained, it is of the utmost importance that the student should soon learn to write down the product rapidly by inspection. This is done by observing in what way the coefficients of the terms in the product arise, and noticing that they result from the combination of the numerical coefficients in the two binomials which are multiplied together; thus (x −8) (x+7)=x2-8x+7x-56 In each of these results we notice that: 1. The product consists of three terms. 2. The first term is the product of the first terms of the two binomial expressions. 3. The third term is the product of the second terms of the two binomial expressions. 4. The middle term has for its coefficient the sum of the numerical quantities (taken with their proper signs) in the second terms of the two binomial expressions. The intermediate step in the work may be omitted, and the products written down at once, as in the following examples : (+2)(+3)= +5c+6. (x −3)(x+4)= x2+x-12. (x+6)(x −9) = x2 — 3x — 54. (x-4y)(x-10y) = x2-14xy+40y2. (x-6y)(x+4y)= x2-2xy - 24y2. By an easy extension of these principles we may write down the product of any two binomials. Thus (2x+3y)(x-y)=2x2+3xy-2xy-3y2 (3x-4y) (2x+y)=6x2-8xy+3xy - 4y2 -6x2-5xy-4y2. 45, These examples will be sufficient, but there are two cases to which it is desirable to direct the student's attention before leaving this part of the subject: their full importance will be developed in later chapters. CASE I. (x+4) ( x − 4) = x2 + 4x − 4x -- 16 = x2-16. (2x+5y) (2x − 5y)=4x2+10xy-10xy – 25y2 CASE II. =4x2 - 25y2. (x+3)2 = (x+3) (x + 3) (3x-4y)2=(3x-4y) (3x — 4y) =9x2-12xy-12xy + 16y2 |