When the quantities to be multiplied together have literal coefficients, proceed as before, putting the sum or difference of the coefficients of the resulting terms into a parenthesis, or under a vinculum, as in addition.

prod. x1—-(a—-b)x3+(p--ab+3)x2+(bp¬-3a)x+3p、

prod. ax1 —(b+ac)x3+(c+bc+u)x2 −(c2 +b)x+c

Ex. 15. Required the continual product of a+ 2x, a-2x, and a2 +4x2.

Multiply a+2x

by a

It may be necessary to observe, that it is usual, in some cases, to write down the quantities that are to be multiplied together, in a parenthesis, or under a vinculum, without performing the whole operation; thus, (a+2x)× (a−2x)X(a2+4x3). This method of representing the multiplication of compound quantities by barely indicating the operation that is to be performed on them, is preferable to that of executing the entire process; particularly when the product of two or more factors

is to be divided by some other quantity; because. in this case, any term that is common to both the divisor and dividend may be more readily suppressed; as will be evident, from various instances, in the following part of the work.

Ex. 16. Required the product of a+b+c by ab+c. Ans. a2ac-b2+c2. Ex. 17. Required the product of x+y+z by x-y-z. Ans. x-y-2yz-22. Ex. 18. Required the product of 1-x+x-x3 by 1+x. Ans. 1-x4. Ex. 19. Multiply a3+3a2b+3ab2 +b3 by a2+ 2ab+b2.

3

Ans. a5+5a b+10a3b2+10a2b2+5ab+b3. Ex. 20. Multiply 4x2y+3xy-1 by 2x2. -X. Ans. 8x+y+2x3y-2x2 -3x2y+x. Ex. 21. Multiply x3+x2y+xy2+y3 by x-y. Ans. x-y. Ex. 22. Multiply 3x3-2a2x2+3a3 by 2x3-3a2 x2+5a Ans.6x6--13a2 x5 +6a2x2 +21a3x3--19a3 x2+15α. Ex. 23. Multiply 2a2-3ax+4x2 by 5u2-6ax --2x2. Ans. 10a4-27a3 x+34a2x2-18ax3-8x4. Ex. 24. Required the continual product of a+x, ̄x, a2 +2ax+x2, and a2-2ax+x2.

Ans, a-3a1 x2 +3a2 x1 —x6. Ex. 25. Required the product of x3-ax2+bx -c, and x2-2x+3.

Ans.

3

x-(a+2)x1+(b+2a+3) x 3 —(c+2b+3a) x2+(2c+36)x-3c. Ex. 26. Required the product of mx3-nx-r and nx-r. Ans. mnx3-(n2+mr)x2 +r3. Ex. 27. Required the product of px2-rx+q and x-rx-g.

Ans. px-(r+pr)x3 + (q+r2 —pq) x2 —q2.

Ex. 28. Multiply 3x2-2xy+5 by x2+2xy-3. Ans. 3x+4x3y-4x2 × (1+y3)+16xy-15. Ex. 29. Multiply a3+3a2b+3ab2+b3 by a3. 3a2b+3ab2-b3.

Ans. a3a4b2 +зa2ba—b®. Ex. 30. Multiply 5a3-4a2b+5ab2-363 by 4a2 -5ab+262.

Ans. 20a5-41a+b+50a3b2 —45a2b3+25aba —6b5.

§ IV. Division of Algebraic Quantities.

30. In the Division of algebraic quantities, the same circumstances are to be taken into consideration as in their multiplication, and consequently the following propositions must be observed.

31. If the sign of the divisor and dividend be like, the sign of the quotient will be +; if unlike, the sign of the quotient will be -.

The reason of this proposition follows immediately from multiplication:

+ab

Thus, if +ax+b=+ab; therefore

=+b:

+a

- ab

+ax-b-ab;

b:

+a

-ab

-ax+b=ab;

+b:

-'a

+ab

-ax-bab;

-b:

-a

32. If the given quantities have coefficients, the coefficient of the quotient will be equal to the coefficient of the dividend divided by that of the divisor.

Thus, 4ab2b, or

4ab

2a.