absolute values (values without regard to their signs) and prefix the common sign to the result.

2. If the numbers have unlike signs, find the difference between their absolute values, and prefix the sign of the greater to the result.

In either case, the result is called the Algebraic Sum.

Where more than two numbers are involved, it is convenient to add two of them, then add their sum to the third and so on; or find the sum of all the positive numbers and the sum of all the negative numbers and add these sums as any two algebraic numbers are added.

ADDITION OF SIMILAR MONOMIALS

28. Since different letters usually stand for different numbers, it is evident that they cannot be combined by addition. Thus, we cannot add a to b, in the ordinary arithmetical way; all we can do is to indicate the addition by connecting them with a plus sign, as, a + b On the other hand, where the letters are the same, as 2a plus 4a, their sum is found by simply adding their coefficients, because, twice a certain number plus four times that number will equal six times that number. If these letters have

different exponents, as 2a2 plus 4a3, they cannot be combined by actual addition, since a2 and a will stand for unequal numbers, just as the different letters a and b represent different numbers.

It follows therefore, that we can only add similar terms, that is, terms having the same letters, with equal exponents, although their coefficients may be different.

1. Add: 2x, 4x, -3x, -5x, 7x.

Adding the coefficients,

2 + 4 + ( −3) + ( −5) + 7 =

Therefore the sum is 5x.

=

+5.

2. Add: 4a2b3, -6a2b3, -9a2b3, -3a2b3.

Adding the coefficients,

4 + ( −6) + ( −9) + ( −3)

Hence, the sum is, -14a3.

= - 14.

Rule for Finding the Sum of Similar Monomials. Add the coefficients and annex to the sum the common letters with their proper exponents.

9. —7z3, 5z3, 6z3, — 10z3.

10. 3xy2, -7xy2, -5xy2, 11xy2, -9xy2.

11. 5abc4abc 7abc + 9abc 2abc.

12. −8x2y2 + 3x2y2 — 5x2y2 — 7x2y2 + 11x2y2. 13. 7√x - 3√x + 8 √ x + √x − 9 √x.

15. 3a + 7a – 5Va – 4Va + Va. 16. bb1b - žb.

17. aa+afa + a. 18. 3c2c2 c2 + ac2 c2.

4,2

20. 3(a+b)

2(a+b)4(a+b)+7(a+b) — 9(a+b).

SUBTRACTION OF ALGEBRAIC NUMBERS

29. To subtract one algebraic number from another, we may apply the rule for removing parentheses.

When a minus sign precedes, a parenthesis may be removed by changing the signs of the terms in the parenthesis.

Hence, the Rule for subtracting one algebraic number from another:

Change the sign of the subtrahend and add the resulting number to the minuend.

This rule can easily be explained if we consider concrete numbers. Suppose a man's assets to be called positive and his debts negative. If a man has $10 and owes $3, he is actually worth only $7. If, however, his debt is removed, he will be worth $10, that is, the subtraction of his debt, the minus $3, is equivalent to an actual addition of plus $3 to his assets.

SUBTRACTION OF SIMILAR MONOMIALS

30. Since any monomial represents some algebraic number, it is evident that the rule used in subtracting one

algebraic number from another, is also used in the subtraction of monomials.

That is, to subtract one monomial from another, change the sign of the subtrahend and then proceed as in addition.

Since only similar monomials can actually be combined by addition, it follows, that only similar monomials can be combined by subtraction. If the monomials are not similar, their subtraction can only be indicated by connecting them by the sign of subtraction. Thus, we can actually subtract 3x from 7x, giving us the remainder 4x, but, the subtraction of 3a from 76 can only be indicated, the difference being written, 76 - 3a.

31. An Integral Expression is an algebraic expression containing no letters in the denominators of any of its terms. Thus, 3x2+5x-2 and ut + at2 are integral expressions.

Note carefully, that while in arithmetic, a number, to be integral must contain no fractions, in algebraic integral expressions, we may have numerical fractions, as in the second expression given above. Also note, that the name, integral expression, refers only to the form of the expression, since even an integral expression may have a fractional value. Thus, the integral expression, 2x — 3y, may have a fractional value if x andy, in which case 2x 3y

=

=

.

=

On the other hand, a fractional

1÷

=

1 X

= 2

ADDITION OF INTEGRAL COMPOUND EXPRESSIONS

-

32. The addition of two integral compound expressions may be indicated by connecting the two expressions by the sign of addition. Thus, to indicate the addition of a + b c and x y+z, we write a + b c+ (x y + z). Removing the parenthesis, we get a + b − c + x − y + z as the sum. If this sum contains any similar monomials, they are combined in the usual way by algebraically adding their coefficients. Thus, the sum of a + 2b - 3c and 2ab+2c equals a + 2b3c + (2a − b + 2c) = 3c + 2a a + 26 3a + b C.

b+2c

=

This operation is most conveniently performed by placing the similar terms in the same column.

This same method is used when more than two expressions are to be added.

Thus, add 8mn2 + 3x2y3 + 5a, 7x2y3 + 3mn2 7x2y3+3a.

Arranging like terms in the same column,

7a, 2mn2

2. 7b2c3d; -8b+ 2c - 11d.

3. 6x4y+ 3t5z; y + 2tz; 2x+5y-4t.

4. 7a5c+ 3x2y; 5x2y - 7b3a; 2c - 4x2y + 3b. 5. 2cbc3a24b2+6ac + 5bc; 4a23bc2b2;

6pq2 - 7p3 + 4p2q + 8q3; 3q3 — 4pq2 — 7p2q + 6p3.