21. From (m + n − x)a + (m −n+x)c subtract

22. Multiply (a + 1)x + (a − 1)y by (a − 1)x + (a + 1)y.

23. Find the sum of 4(a+b+1), 3x(a+b+1), −2b(a+x+1), and (7 — 3x) (a+b+1).

24. Simplify

3a +(2a+1)(2a-3) - (4a-1)(a + 1) + (2a - 1)2(4a-1). 25. Find the continued product of (x+a) (x + b) (x + c).

26. Simplify (a− x) (y − a) + (x − y) (a − x) + (y − a)(x − y).

27. Multiply a2 + a + } by a2 - 3a +1.

28. Multiply 1.2x23 +1.4x2+1.5x+1.6 by 0.5 x2- 0.1 x +0.2. 29. Simplify 2[3 a {a+2(3a-2a-1)}-2(3a- a — 5)]. 30. Find the product of (x2 + x + 2)(x2 − x + 2)(x1 − x2 + 4). 31. What must be added to (a + a + 1)2 to give (a — x − 1)2? 32. Multiply x2 + xy + y2 by x2 - xy + 3y2. 33. Show that 4 m2 + (m + 1)2 + 1 = (m −1)2 + (2 m + 1)2. 34. Multiply cx2 + dx + 1 by ax+2.

35. Multiply a2 + b2 + c2+2 ab- ac-bc by a+b+c. 36. Show that

(c2d2 + cd+1)2+(c2d2 — cd+1)2 = 2 (c1d1 + 3 c2d2 + 1).

37. Simplify 2[3 x − 2 { 2 x − 2 (x −3x+1)} +9x+4].

38. Multiply 3 + } x2 − 1 x + 1 by } x2 − } x + 1,

39. Multiply mx2 - nx+1 by ax-1.

40. Simplify

(ab+a+b+1)2 - (abb + a1)2 - 4 (ab2 + a + b)

CHAPTER V

DIVISION. REVIEW

72. Division is the process of finding one of two factors when their product and one of the factors are given. The dividend is the given product; the divisor, the given factor; and the quotient, the factor to be found.

THE NUMBER PRINCIPLE OF DIVISION

73. The Law of Distribution.

The quotient of a polynomial by

a monomial equals the sum of the quotients obtained by dividing each term of the polynomial by the monomial.

74. The significance of this law is that the divisor is distributed as a divisor of each term of the dividend.

75. The process of division being the inverse of the process of multiplication, the laws governing signs, coefficients, and exponents in multiplication form, when inverted, the corresponding laws for division.

Therefore, from (1) and (4), and from (2) and (3):

76. Like signs in division give a positive result.

77. Unlike signs in division give a negative result.

COEFFICIENTS IN DIVISION

78. The coefficient of a quotient of two algebraic expressions is obtained by dividing the coefficient of the dividend by the coefficient of the divisor.

In general, therefore, we have the following:

If m and n are any positive integers and m is greater than n

ama xa xa to m factors,

...

Hence, the statement of the second index law is made as follows:

79. The quotient of two powers of a given factor is a power whose exponent is the exponent of the dividend minus the exponent of the divisor.

80. Any quantity with a zero exponent equals 1.

66

We would obtain the same result by the old method of saying "3 in x3 once"; and the quotient "1" obtained in this way is a factor of the complete quotient.

DIVISION OF A MONOMIAL BY A MONOMIAL

By application of the laws established for signs, coefficients, and exponents, we obtain a process for the division of a monɔ mial by a monomial.

35 a5x3y1 = 5 a5-3x3-3y4-2 = - 5 a2y2. Result.

Hence, to divide a monomial by a monomial:

81. Divide the coefficient of the dividend by the coefficient of the divisor, annexing to the result the literal factors, each with an exponent equal to its exponent in the dividend minus its exponent in the divisor.

4a)-12 a3-3x2)15 x3y 11 xy)-44x3y2 -7 a3b)- 28 a3b3

DIVISION OF A POLYNOMIAL BY A MONOMIAL

82. From the Law of Distribution (Art. 75), and the principles governing the division of monomials (Art. 81), we obtain a process of division when the dividend is a polynomial and the divisor a monomial.

In practice the usual form of expression is a)ax +ay + az

Illustration:

x + y + z

Quotient.

1. Divide — 6 m1n + 10 m3n2 — 14 m2n3 + 4 mn1 by — 2 mn.

— 2 mn) — 6 m1n + 10 m3n2 — 14 m2n3 + 4 mn1

3 m3

5 m2n + 7 mn2 - 2 n3 Result.

Each term of the quotient is obtained by applying the principle of Art. 81, for in the division of each term by the monomial divisor we have a division of a monomial by a monomial.

Hence, to divide a polynomial by a monomial:

83. Divide separately each term of the polynomial by the monomial and connect the terms of the resulting polynomial with the proper signs.