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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.
Mul. 2 -ax+p
by x2 +bx+3
t-3x2 — 30x +3p
prod. x*--(a-b)x3+(p-ab+3)x2 +(bp--3a)x +3p
prod. ax4 -(6+ac)x3 +(c+bc+a)x2 -(02 +b)x toc
Ex. 15. Required the continual product of at 2.0, a--2, and a +4.co.
by a - 2.0
Total product a*
-16x4 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) X(a-2x) X(ao +-4."). 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 ah+C.
Ans. a” +-2ac-62 +ca. Ex. 17. Required the product of x+y+z by y-2.
Ans. x2 —Yo — 2yz - zo. Ex. 18. Required the product of 1-*+x2-23 by 1+x.
Ans. a5 + 5a4b+10a3b2 +10aob3 +- 5abi +65.
Ans. 8x*y+ 2x’y-2.? --3.co y + x.
Ex. 23. Multiply 2a?—3ax +4x2 by 541—6ax -- 2x2. Ans. 10a4 - 27a3x + 34a2 x 2 - 18ax 3 -- 8 v4.
Ex. 24. Required the continual product of a+x.
Ans, ao - 3a*r? +3ac-.
22+(2+36)2--3c. Ex. 26. Required the product of mx?-n&—1 and nour.
Ans, mnx3--(na + mr)to 12. Ex. 27. Required the product of pica-r*+ and 22.
Ans. px -(r+pr)x3+(2+ord--p9)-q
Ex. 28. Multiply 3x2 – 2xy +5 by xo + 2xy--3.
Ans. 3x4 + 4xy--4xco X(1+y)+16xy-15. Ex. 29. Multiply a3 +3a2b+3ab2 +63 by a3. 3a2b +3ab2-63. Ans.a-3a112 +39-64-66
Ex. 30. Multiply 5a” -4a2b+ 5ab2 — 363 by 4a2 - 5ab +362. Ans. 20a5-41a+b+-50a312 -- 45a263 +25ab4-655.
§ 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 t; if unlike, the sign of the quotient will be
The reason of this proposition follows immediately from multiplication :
tab Thus, if tax+b= tab; therefore
-a X-b= tab;
82. If the given quantities have coefficients, the con efficient of the quotient will be equal to the coefficient of the dividend divided by that of the divisor.
4ab Thus, 4ab-;-2b, or 20.