a

(a+b)?=a? +2ab+62

(1) (a−b)?=a2–2ab+62

(2) (a+b): +(a–6)2=2a2 +262

(3) f(a+b)+(a−b)=a

(4) (a+b)-(a−b)=

(5) Expression (1) is read, “ the square of the sum of a and b is equal to the square of a, plus twice the product of a and b, plus the square of b."

Expression (2) is read, “the square of a diminishd by b is equal to the square of a, minus twice the product of a and b, plus the square of b."

The student should read the remaining expressions for himself, and should also form other expressions, which he may in like manner translate into common language. He should also substitute particular values for a and b, in the above expressions, and see if the results on both sides of the equations are identical. Thus, the above expressions become, when

a=2, and b=1
(2+1)"=4+4+1=9

(1) (2-1)2=4-4+1=1

(2) (2+1)2+(2-1)2=8+2=10

(3) (2+1)+(2-1)=2

(4) (2+1)-3(2-1)=1

(5) If a=ì, and b=} they will become (1+3)=1+3+1=35

(1) (1-3)=1-1+1=36

(2) (1+3)2+(1-3)?=1+3=i}

(3) (1+3)+(1-3)= *(1+3)-3(3-})=}

(5) In this way the student should be exercised, until he becomes familiar with the nature of algebraic expressions.

2

(4)

2

ADDITION.

(23.) ADDITION, in Algebra, is finding the simplest expression for several algebraic quantities, connected by + or -. Suppose we wish to find the sum of

3a2b+7a2b-10a2b+4a2b-5a2b-2a2b.

We first seek the sum of the positive quantities, by placing them under each other as in arithmetical addition, thus,

Proceeding in the same way with the negative terms, we

CASE. I. (24.) When the quantities are alike but have unlike signs, we have this

RULE.

I. Place the different terms under each other, add the coefficients of the positive quantities into one sum, and the coefficients of the negative quantities into another.

II. Subtract the LESS from the GREATER,

III. Prefix the sign of the greater sum to the remainder, and annex the common letters.

EXAMPLES.

1. What is the sum of

2abx-7 abr-2abx+12abx+ab.x-3abx?

2abx 12abr

abr

15abx= sum of positive terms.

-7abx -2abc -3abc

-12abx= sum of negative terms.

Therefore 15abx-12abx=3abx=sum total.
2. What is the sum of
7amn-3amn+2amn+hamn-10amn?

Ans. amn.

3. What is the sum of

49axy-37axy-10axy+100axy-7axy+4axy?

4. What is the sum of

Ans. 99axy.

3√ax+7√ax-5 √ax-3 √ax+10 √ax-4√ax?

(25.) When both quantities and signs are unlike, or some like and others unlike.

RULE.

I. Find the sum of the like terms as in Case I.

II. Then write the sums one after another, with their proper signs.

EXAMPLES.

1. What is the sum of

3ax-2ab+4xy-2ax+3xy+7ab-2xy+6ax?

3ax

-2ax

6ax

7ax=sum of the terms containing ax.

-2ab
7ab

5ab sum of the terms containing ab.

4xy 3.xy -2xy

5xy=sum of the terms containing xy.

Therefore 7ax+5ab+5xy=total sum.

2. What is the sum of

2a2x-3ax2+2ab-7a2x+4ax2-8ab-6a2x+10ax2.

+12ab?

Ans.

2a2x-3ax2+2ab

-7a2x+4ax2- 8ab

-6a2x+10ax2+12ab

-11a2x+11ax2+ 6ab

3

3a2b37ab4+5axy -7a2b32ab+-axy 8a2b3+ ab1-7axy a2b3-10ab+3axy