Division of a Polynomial by a Monomial.

9. The quotient resulting from the division of a polynomial by a monomial may be obtained as a direct application of the Distributive Law for division. That is, since

(a + b−c)÷d=a÷d+b÷d-c÷d,

it follows that we may divide each term of the polynomial dividend by the monomial divisor and write the algebraic sum of the resulting partial quotients.

Ex. 1. Divide

15 ab2 10 a2b3 + 5 ab5 by 5b2.

(15 a1b2 — 10 a2b3 + 5 ab5) ÷ 5 b2 = 15 ab2 ÷ 5 b2 – 10 a2b3 ÷ 5 b2 + 5 ab3 ÷ 5 b2

3a42a2b + ab3.

Check. Let a = 3, b = 2. (4860-720 +480) ÷ 20 = 243 — 36 + 24

231231.

1. 2a) 6 ab + 8 ac.

2. 3b) 12 bc + 18 bx.

MENTAL EXERCISE VIII. 3

Perform the following indicated divisions:

3. 6 d) 30 dxy + 42 d.

12.

4 c1

32. ({ a23c2 + } a3b2c3 + } a2b3c3) ÷ 3a2b2c2.

34. (m3n3w } } m3 n*w3 — + § m*n3w3) ÷ 5 m3n3w3.

35. (a*bd3 — § a3b*d — § a2b2d2) ÷ (— } a2bd).

36. [3(a + b)3 + 9(a + b)* + 6(a + b)3] ÷ 3(a + b)2.
37. [5(x + y)° — 15(x + y)3 — 30(x + y)*] ÷ 5( x + y)3.
38. [4(b − d)3 + 20(b − d)3 + 16(b − d)'] ÷ 4(b — d)3.
39. [6(g2 — h2) + 18(g − h)2 + 12(g − h)3] ÷ 6(g — h).
40. (am++a+b+a+1) ÷ a.

47. (a1x+y+a3 x + 2y + a2 x + 3y + a 2 + + y) ÷ ax+”.

m+2n
2

49. (a2m+13+ a2m+25 + a2m+817) ÷ a2mb2.

Division of One Polynomial by Another.

10. An expressed quotient is called a fraction; the dividend is called the numerator and the divisor the denominator.

11. The process by which the division of one polynomial by another is performed, may be made to depend upon the following

Fundamental Principle: The quotient obtained by dividing an integral function of a by an integral function of x which is of degree not higher than that of the dividend can be transformed into the sum of an integral function of x (the integral quotient) and a fraction the

numerator of which is the remainder resulting from the division, and the denominator of which is the given divisor.

The degree of the integral quotient obtained by the above Division Transformation is equal to the excess of the degree of the dividend over that of the divisor.

The degree of the remainder which is used as the numerator of the fractional part of the transformed function is accordingly less than the degree of the divisor which is used as a denominator.

(The following proof may be omitted when the chapter is read for the first time.)

Let the dividend D and the divisor d be integral functions of some letter z, the degree of the divisor being not higher than that of the dividend with reference to x. Letting

stand for the integral part of the quotient and R for the remainder, we have

= Q + (D ÷ d
=Q+ (D ÷ d

Q). Commutative and Associative Laws. Q) xdd. Since x d d = 1. =Q+(D÷dxd-Qxd)÷d. Distributive Law. =Q+ (D - Qd) d. Since d x d 1.

=Q+R÷d. Since D

Qd is by definition the same as R.

Division of one Polynomial by Another.

DEVELOPMENT OF THE PROCESS

12. In order to clearly understand the process of division, it is well to obtain the product of two given integral polynomials, and then, after having examined carefully the manner in which it is built up, to reverse certain of the steps and processes. Then starting with the reduced product as a dividend, and one of the factors, say the multiplicand, as a divisor, we may find the other, the multiplier, which we shall now call the quotient.

13. For convenience, we shall select two polynomials in which no powers are missing, and shall arrange them according to descending powers of the same letter, say a.

Multiply a3 +3a2b + 3 ab2 + b3 by a2 + 2 ab + b2.

Reduced Product

Partial

Products.

Divisor. Quotient.

+2ab+6 a3b2 +

+ a3b2 +

3a2b3 + 3 aba + b5

a5 +5 ab + 10 a3b2 + 10 a2b3 + 5 ab1 + b5 Dividend. 14. Observe that the number of terms in each horizontal row of partial products corresponds to the number of terms in the multiplicand, and there are as many rows as there are separate terms in the multiplier. The degree of the first term of each row with reference to the letter of arrangement is higher than that of any following

term.

15. If now we interchange the given polynomials and use the first as a multiplier, and the second as a multiplicand, we shall obtain the same reduced product and the same partial products as before; but their orders of arrangement will be different, as in Form II.

16. Each horizontal row of partial products in Form II corresponds to an oblique or diagonal row containing the same terms in Form I, and each row in Form I has a corresponding oblique or diagonal row in Form II.

E. g. The first horizontal row in Form II, a5 + 2 ab + ab2, appears as the first diagonal row a + 2 ab + a3b2 in Form I (§ 13). Also, the terms'