17. From 2 d + 11 a + 10 b 5 c take 2 c + 5 a 3b; and show that the result is numerically correct when a = 1, b = 3, c = 5, d = 7. 18. If a = 1, b = - 3, C=- - 5, d= 0, find the value of a2+262+3 c2 + 4 d2. If a = 3, b = value of: 4, c9, and 2 s = a+b+c, find the 21. s2(sa) (s—b) — (s—b) (s—c) — (s — c) (s — a). 22. If x = a+2b-3c, y=b+2c-3a, z=c+2a-3b, show that x+y+z=0. 23. If x = a -2b+3c, y=b-2c+3a, z=c2a+3b, show that x+y+z=2a+2 b + 2 c. 24. What must be added to x2+5y2+322 in order that the sum may be 2 y2 — z2? 25. What must be added to 5 a3 - 7 a2b + 3 ab2 in order that the sum may be a3 - 2 a2b — 2 ab2 + b3 ? If E=5a3+3a2b-263, F3 a3 - 7 a3b-b3, find the simplest expression for: 26. E+F-G-H}. 27. E-{F+G— H}. 28. EF-G+H}. 29. E-F- (GH)}. {F Hab-2 a3-3 b3, 30. E+G (FH). 31. F-H-(E + G). 32. H-E-(F — G). 33. FG (EH). 34. From 2x-2y-z take 3y+2x2-22, and from 35. Take the sum of a2c 2 a3 + 2 ac2 and as. ac2 — a2c from a3 2 a2c + 3 ac2. CHAPTER V. MULTIPLICATION AND DIVISION. Multiplication of Compound Expressions. 96. Degree of a Term. A term that has one letter is said to be of the first degree; a term that is the product of two letters is said to be of the second degree; and so on. 97. Degree of a Compound Expression. The degree of a compound expression is the degree of that term of the expression which is of the highest degree. Thus, a2x2 + bx + c is of the fourth degree, since a2x2 is of the fourth degree. 98. When all the terms of a compound expression are of the same degree, the expression is said to be homogeneous. Thus, x+3x2y + 3 xy2 + y3 is a homogeneous expression, every term being of the third degree. 99. Dominant Letter. If there is one letter in an expression of more importance than the rest, it is called the dominant letter; and the degree of the expression is called by the degree of the dominant letter. Thus, a2x2 + bx + c is of the second degree in x. 100. Arrangement of a Compound Expression. A compound expression is said to be arranged according to the powers of some letter when the exponents of that letter descend or ascend, from left to right, in the order of magnitude. Thus, 3 ax3 — 4 bx2 − 6 ax + 8 b is arranged according to the descending powers of x; and 8b-6 ax 4 bx2 + 3 ax3 is arranged according to the ascending powers of x. Multiply each term of the polynomial by the monomial, and connect the partial products with their proper signs. Find the product of ab + ac-be and abc. ab + ac bc abc a2b2c + a2bc2 ab2c2 NOTE. We multiply ab, the first term of the multiplicand, by abc, and work to the right. 3. ab 5. x-y-zby -3xy. 7. a2 + b2 - c2 by a3bc2. 3 a3bc2 by 2 ab2c. 10. xyz2+x2y3z by - xyz. 13. ax-5a2x2 + ax3 + 2 x1 by ax3y. 14. 9a3 x b2 - 4 a2b3 — b5 by - 3 ab*. 15. 3a3-2x2y-7xy2+ y3 by - 5 x2y. Multiplication of Polynomials by Polynomials. If we have m + n + p to be multiplied by a + b + c, we may substitute M for the multiplicand m +n+p. Then (a+b+c) MaM+bM+cM. (§ 41) If now we substitute for M its value m +n+p, we have aM+bM+ cM = a (m + n + p) + b (m +n+p)+c (m+n+p) = am+an+ap + bm + bn + bp + cm + cn + cp. 102. To Find the Product of Two Polynomials, therefore, Multiply every term of the multiplicand by each term of the multiplier, and add the partial products. In multiplying polynomials, it is a convenient arrangement to write the multiplier under the multiplicand, and place like terms of the partial products in columns. We multiply 5 a, the first term of the multiplicand, by 3 a, the first term of the multiplier, and obtain 15 a2; then we multiply - 66, the second term of the multiplicand, by 3 a, the first term of the multiplier, and obtain - 18 ab. The first line of partial products is 15 a2 18 ab. In multiplying by 4b, we obtain for a second line of partial products — 20 ab + 24 62, and this is put one place to the right, so that the like terms - 18 ab and 20 ab may stand in the same column. We then add the coefficients of the like terms, and obtain the complete product in its simplest form. 2. Multiply 4x + 3 + 5 x2 - 6 x3 by 4 - 6 x2 - 5 x. Arrange both multiplicand and multiplier according to the ascending powers of x. 3. Multiply 1+2x+x-3x2 by x3-2-2x. Arrange according to the descending powers of x. NOTE. The student should observe that, with a view to bringing like terms of the partial products in columns, the terms of the multiplicand and multiplier are arranged in the same order. |