2. What is the square of 2x2-5 ?

The square of the first term is 4x, twice the product of the twe terms is 20x3, and the square of the second term is 25; hence by the second formula,

(2x-5)=4x-20x2+25, Ans.

3. What is the product of 5x+y' and 5x-y'?

The square of 5x is 25x2, and the square of y' is y'; hence by the third formula,

(5x+y3) (5x—y3)=25x3—y1, Ans.

4. What is the square of c+m?

5. What is the square of x―y?

6. What is the product of x+y and x-y?

7. What is the square of 3x+4y?

8. What is the square of 5c3-2cd?

Ans. c+2cm+m3.

Ans. x-2xy+y'.

Ans. x2-y'.

Ans. 9x+2+x3y+16y2.

Ans. 25c-20c'd+4c❜d".

9. What is the product of 4x+3yz and 4x3-3yz?

Ans. 16z-9y'z'.

10. What is the square of 3a x+2ay?

Ans. 9ax+12a'xy+4a3y3.
Ans. x+2x+1.

11. What is the square of x+1?

square of 2x2-1?

12. What is the
13. What is the product of m+1 and m-1?

14. What is the square of z3-30 ?

Ans. 4z*—4z2+1.

Ans. m2-1.

Ans. z-60%+900.

15. What is the product of 3ab+d and 3a'b-d'?

16. What is the square of x-ty?

17. What is the square of 2c++?

square

of x+y"?

Ans. 9a'b'-d® Ans. x2—xy+ty3.

Ans. 4c+2c+4.

2m

Ans. x+2xy"+y2".

18. What is the 19. What is the product of x+y" and x-y"? Ans. x2m-yan.

2m

2n

71. The binomial square occurs so frequently in algebraic operations, that it is important for the student to be perfectly familiar with its form. The higher powers of any binomial may be obtained by actual multiplication. The 3d, 4th, and 5th powers, however, may sometimes be easily written, without actual multiplication, by means of the formulas which follow:

1. (a+b)'=a'+3a2b+3ab2+b'.

2. (a-b)'=a'-3a2b+3ab’—b'.

3. (a+b)'=a*+4a3b+6a2b2+4ab'+b*.

4. (a—b)'=a*—4a3b+6a2b2—4ab'+bʻ.

5. (a+b)=a+5ab+10ab2+10a b'+5ab'+6°. 6. (a-b)-a-5ab+10a b'-10a l'+5al-bo.

Let the pupil verify the above by actual multiplication.

72. A polynomial is said to be arranged according to the descending powers of any letter, when the terms are so placed that the exponents of this letter diminish from left to right throughout all the terms that contain it. Thus, the polynomial

x-4x+2x-x+7

is arranged according to the descending powers of x.

73. A polynomial is said to be arranged according to the ascending powers of any letter, when the terms are so placed that the exponents of this letter increase from left to right throughout the terms that contain it. Thus, the polynomial

d-ax+cx3-bx*

is arranged according to the ascending powers of x.

74. A term or quantity is said to be independent of any letter, when it does not contain that letter.

75. The product of two polynomials has certain special properwhich may be stated as follows:

ties,

1. If both polynomials are arranged according to the descending powers of the same letter, then the first term obtained in the partial products will contain a higher power of this letter than any of the other terms; and as this term can not be reduced with any of the others, it will form the first term of the entire product.

2. If both polynomials are arranged according to the ascending powers of the same letter, then the last term obtained in the partial products will contain a higher power of this letter than any of the other terms; and as this term can not be reduced with any of the others, it will form the last term of the entire product.

3. If both polynomials are homogeneous, then the product will khomogeneous; and the degree of any term will be expressed by the sum of the indices denoting the degrees of its two factors.

DIVISION.

76. Division, in Algebra, is the process of finding how many times one quantity, called the divisor, is contained in another quantity, called the dividend; the result of division is called the quotient.

It follows, therefore, that the quotient must be a quantity which multiplied by the divisor, will produce the dividend. Thus, reversing the process of multiplication, we have,

abc÷a bc, because bc Xa=abc

77. It was shown in the multiplication of monomials, (66), that the coefficient of the product is found by multiplying together the coefficients of the factors; and that the exponent of any letter in the product is found by adding together the exponents of this letter in the factors. Hence, in division,

1. The coefficient of the quotient must be found by dividing the coefficient of the dividend by that of the divisor; and

2.--The exponent of any letter in the quotient must be found by subtracting the exponent of this letter in the divisor from its exponent in the dividend. Thus,

24a*÷6a3—24a5—3=4a3

It was shown in multiplication, (66), that when two factors have like signs, their product is positive; and that when two factors have unlike signs, their product is negative. In division, therefore, when the dividend is positive, the quotient must have the same sign as the divisor; and when the dividend is negative, the quotient must have the sign unlike that of the divisor. And there will be four cases, with results as follows:

3.—If the dividend and divisor have like signs, the quotient will be positive; but if the dividend and divisor have unlike signs, the quotient will be negative.

CASE I.

78. When the divisor is a monomial.

From the principles already given we have the following
RULE. To divide one monomial by another;-

I. Divide the coefficient of the dividend by the coefficient of the divisor, for a new coefficient.

II. To this result annex the letters of the dividend, with the exponent of each diminished by the exponent of the same letter in the divisor, suppressing all letters whose exponents become zero.

III. If the signs of terms are alike, prefix the plus sign to the quotient; if they are unlike, prefix the minus sign.

To divide a polynomial by a monomial;—

Divide each term of the dividend separately, and connect the quotients by their proper signs.

NOTE.-It may happen that the dividend will not exactly contain the divisor; in this case the division may be indicated, by writing the dividend above a horizontal line, and the divisor below, in the form of a fraction. The result thus obtained may be simplified, by suppressing all the factors common to the two terms; thus,

4x2yz2+6x2 y2z=

4x2yz2 22
6x2y22 3y

But as this process is essentially a case of reduction of fractions, we shall omit such examples till the subject of fractions is reached.

11. Divide 34xy" by —17xy. 11. Divide (a—-c)° by (a—c)3.

13. Divide 35(x+y)3 by 5(x+y).

Ans. -2x-17-1.
Ans (a-c).

14. Divide 12m'd(c-x')' by 3md (c-x2)'.
15. Divide 3bcd+12bcx-9l3c by 3bc.
16. Divide 15a2bc-15acx+5ad'c by -5ac.
17. Divide 10x3-15x-25x by 5x.

Ans. 7(x+y)2.

Ans. d+4x-36.

Ans. 2x-3x-5.

18. Divide 153—45x1+10x3—105x2 by 5x3. 19. Divide ac-am-1c2+a2c3-a-c+am-c' by ac. Ans. am-1—am-3c+am¬3 c2 —am—^c3+aTM¬3c*.

20. Divide 3m2(a—b)2—3m(a—b) by 3(a—b).

m

Ans, am❜-bm2—m.

21. Divide 7a(3m—2a)—(3m—2a)3 by (3m—2a).

Ans. 9a-3m.

CASE II.

79. When the divisor is a polynomial.

Suppose both dividend and divisor to be arranged according to the descending powers of some letter. Then it follows, from (75, 1), that the first term of the dividend must be the product of the first term of the divisor by the first term of the quotient similarly arranged. We can therefore obtain this term of the quotient, by simply dividing the first term of the dividend by the first term of the divisor, thus arranged. The operation may then be continued in the manner of long division in Arithmetic; each remainder being treated as a new dividend, and arranged as the first. 1. Divide 6a+a3b—20a3b2+17ab3-46* by 2a3—3ab+b2.