NOTE. The above might have been done by writing all the four quantities under each other, like an account in addition, observing to change the sign of every quantity in the subtrahend.

(a+1)(a+2)=a2+ 3a + 2
(a+3)(a+4)=a2 + 7a +12
a1+ 3a3 + 2a2

7a3+12a2+14a

12+a+36a+24

a1+10a3 +35a2+50a+24

14th.

(a2+ax+x2)(a2—ax+x2)=a1+a2x2 + x1 See Ex.

12th and (a-x)(α+x)

x+4x3y-60x2 y211+x2-2xy +28

16th.
3x)6x-3x+12x2+14x
2x3. x2 + 4x + 42

x+y)x5+ y3 (xa—x3y+x2y?—xy3 +y4

x2+x+y

-x1y+ y5

-x1y—x3 y2

x2+y3)x5+x2y2+x3y3+y3 (x3 + y2

79

21st.

x-1)x-px1+qx3—qx2+px—1(x*—(p—1)x3+(q—p+1)x®—

x5. -X4

−(p—1)x1+qx3
-(p—1)x2+(p—1)x3

(p-1)x+1

(q-p+1)x3-qx2
(q-p+1)x3-(g-p+1)x2

—(p—1)x2+px
—(p—1)x2+(p−1)x

Since the multiplicand is the same in all the three sums here given, the sum of the products may be obtained by multiplying it by the sum of the multipliers, which is a+b+c.

a+b+c

a+b+c

a2+ab+ac

ab +b2+bc

ac +be+c2

a2+2ab+2ac+b2+2bc+c2

or a2+b2+c2+2ab+2ac+2bc

REMARK. From the above it is evident, that the square of the sum of three quantities, is equal to the sum of the squares of these

quantities, together with twice the product of every pair of the quantities. If some of the quantities had been minus, the rule would still hold, only when the quantities multiplied together had unlike signs, the sign of the product would be minus. It is universally true, that the square of the sum or difference of any number of quantities is equal to the sum of the squares of the quantities, together with twice the product of every pair of the quantities taken, with their proper signs, according to the rules of multiplication.

24th.

(a+x)(a+x)=a2+2ax+x2. Alg. art. 29.
(a—x)(a—x)=a2-2ax+x2. Alg. art. 30.
(a+x)(a—x)=a2 -2. Alg. art. 31.

(a+x)(a+x)=a2+2ax+x2. Alg. art. 29.
(a—x) (a—x)=a2-2ax+x2. Alg. art. 30.
Sum =2a2 +2x2.

Difference = 4ax.

REMARK. The above proves the following theorems:-The square of the sum, together with the square of the difference of two quantities, is equal to twice the sum of their squares; and the square of the sum, diminished by the square of the difference of two quantities, is equal to four times their product.

26th.

By the remark on question 23d, it is plain, that
(a+b+c)2=a+b2+c2+2ab+2ac+2bc,

and (a+b-c)=a2+b2+c2+2ab-2ac-2bc;
=2a2+262+2c2+4ab,

... the sum

a2+2ab+2ac +b2+2bc +c2-d2

or a2+b2+c2-d2+2ab+2ac+2bc (A)

a+b=c+d