Add the following, with reference to the similar parts. In case two unlike terms contain the same letter or factor, we may perform the addition with reference to this letter as a summand, regarding the remaining factors as coefficients.

Ex. 69. Find the sum of az and yz.

Regarding x and y as coefficients of z, the sum may be expressed by writing the sum of x and y as a coefficient of z, as follows: xz + yz = (x + y)z, or in vertical arrangement, xz

Expressions which contain a common binomial or a common polynomial factor may be added with reference to this common factor.

We may regard the factors which are not common as being coefficients with reference to the common factors and, finding their sum as positive and negative numbers, prefix this as a coefficient to the common factor.

Ex. 90. Find the sum of a(x + y) and b(x + y).

Regarding a and b as coefficients of the common factors of the two given terms, we may write their sum as a coefficient to the common part. a(x + y) + b(x + y) = (a + b) (x + y).

Ex. 91. Find the sum of 3(x2 — y2), 4 a (x2 — y2), and — 2 b (x2 — y2). Adding the coefficients 3, 4 a, and 2b with respect to the common factor x2 y2, we have

3(x2- y2) + 4 a(x2 — y2) — 2 b(x2 — y2) = (3 + 4 a − 2b) (x2 — y2).

27. The algebraic difference of two numbers or quantities may always be indicated by writing the subtrahend after the minuend, separating the two by the sign of operation for subtraction —, each being preceded by its proper quality sign.

From the principles affecting operations with positive and negative numbers it appears that instead of a subtraction of a number we may substitute the addition of a number equal to it in absolute value but of opposite quality.

That is, to subtract a given number or term we change its sign of quality and then proceed as in addition.

As in the case of addition, this collecting of several numbers or quantities into one algebraic expression is what in algebra is called subtraction. The resulting expression is called the difference. Ex. 1. From 5 x subtract 2x.

Indicating the process by the horizontal arrangement, we have

5x-2x=(5 — 2)x = 3x.

5x

When employing the vertical arrangement, 2x, instead of actually altering the sign of quality of the subtrahend 2x, we should make the change mentally when performing the operation, and write the result immediately underneath as in arithmetic.

Ex. 2. From - 3 y subtract 7 y.

Using the horizontal arrangement, we have

-3y(+7y)=-3y-7y = (-3-7)y=-10y.

28. Caution. The student should be careful, when employing the vertical arrangement for subtraction, not to actually change the sign of the subtrahend on paper. Such a change of sign is confusing when the work is reviewed.

29. It is customary, whenever possible, to so arrange an expression as to have the first term preceded by a positive rather than a negative sign.

E. g. We should consider written a b.

ba to be arranged in better order if

MENTAL EXERCISE. VI. 2

Perform the following indicated subtractions :

In case two unlike terms contain the same letter or factor, we may perform the subtraction with reference to this letter or factor, regarding the remaining factors as coefficients.

Ex. 65. From ax subtract bx.

Since a and b are the coefficients of x, we may express the difference by writing the algebraic difference (a - b) as a coefficient of the common letter z, or (a - b) x.

Perform the following subtractions with reference to similar parts:

The difference between two compound expressions which contain the same polynomial factor may be found with reference to this polynomial factor.

Ex. 86. From a (x + y) subtract −b (x + y).

Regarding a and - b as coefficients of the binomial factor (x + y) we may find the difference by prefixing the algebraic difference of a and — b, as positive and negative numbers, as a coefficient to the common factor (x+y) as follows:

a (x + y) -b (x + y)

(a + b) (x + y)

30. A polynomial is said to be reduced when its like or similar terms have all been combined, as far as possible; that is, when it contains no similar terms.

Ex. 1. Reduce 3x + 2y + 5%

We have

2x-3y+7% + 4y to simplest form.

3x+2y+5% -2x-3y+7 z+4y = x + 3y+ 12 z.

EXERCISE VI. 3

Reduce each of the following polynomials to simplest form:

1. 2a4b+6ca5b-2 c.

2. 5x7y-2-4x-6y+z.

3. 11a2d+3b+4d-b+a.

4. 12ab+d+3c+2b-2c-a+3b.