SUBTRACTION.

(33.) Algebraic SUBTRACTION consists in finding the difference between the two quantities; that is, in finding what quantity added to the quantity subtracted, will produce that from which the subtraction is made.

Thus the difference found by subtracting 3a from 5a is 2a, because 2a added to 3a produces 5a, (27).

But the difference found by subtracting -3a from 5a is 8a; because 8a must be added to -3a to produce 5a, (29).

In the second example, observe that subtracting -3a has an effect contrary to that of subtracting 3a in the first example, (25); the Subtracting of a negative quantity being the same as the Adding of an equal positive quantity.

Subtraction of Monomials.

(34.) A monomial is subtracted from another quantity, by changing the sign of the monomial, and then adding it, algebraically, to the other quantity.

Thus 5a subtracted from 2a gives -3a, because -3a must be added to 5a, to produce 2a, (29); and this difference, -3a, will be found by changing 5a to -5a, and adding the -5a to the 2.a

What difference will be found by subtracting 5a from 9a? and why is that difference true? By subtracting 9a from 5a?

What difference will be found by subtracting 3ab from 4ab? and why is that difference true? By subtracting -5x from -10x? and why is that difference true? By subtracting 7ax from -3ax.

(35.) When the two quantities are dissimilar, a monomial is subtracted by changing its sign, and then connecting it with the other quantity, in a Polynomial.

Thus 2x subtracted from 3a gives 3a-2x; and -2x subtracted from 3a gives 3a+2x, (34), (30).

What difference will be found by subtracting 2a from 3x? By subtracting -36 from 5ax? By subtracting ax from 106? By subtracting 4y from -9x 2?

From the preceding we derive

RULE II.

(36.) For the Subtraction of Algebraic Quantities.

to

Change the sign of each term to be subtracted, + to and +, or conceive these signs to be changed, and then proceed as in Algebraic Addition.

EXAMPLE.

To subtract a2+2ab-3x-z from 3a2-5ab+5x—y.

3a2-5ab+5x-y

a2+2ab-3x-z

2a2-7ab+8x-y+z

Having set the polynomial which is to be subtracted under the one from which the subtraction is to be made, we conceive each term in the second line to have its sign changed; then -a2 added to 3a2 makes 2a2; -2ab added to -5ab makes -7ab; and 3x added to 5x makes 8x. These results, together with the -y and +z, are connected in a polynomial.

EXERCISES.

From 4ab+3c2-xy, subtract 2ab-c2+3xy-2z.

Ans. 2ab+4c2—4xy+2z.

2 From 5a2+3bc-cd+3x, subtract 2a2-bc+3cd.

Ans. 3a2+4bc-4cd+3x.

3 From 7a2b-abc+4xy, subtract a2b+5abc-xy+mz.

Ans. 6a2b-Cabc+5xy—mz.

4 From 8a3-4a2b—2bc+10, subtract 3a3+a2b−5.

Ans. 5a3-5a2b-2bc+15. 4. From 362c+abx+2xy2, subtract b2c-3abx+2xy2-3.

Ans. 262c+4abx+3.

6. From 4a2c-3c2b+6y+20, subtract 3c2b+6y+3by.

Ans. 4a2c-6c2b+20—3by..

7. From -3a+562-7xy, subtract 362+2a-xy+bc.

Ans. -5a+262-6xy-bc.

8. From 2ab+3bc-2ax2+15, subtract 7ab+3bc-10.

Ans. -5ab-2ax2+25.

9. From 5ax-3bc+262, subtract 6ax-bc-362+y2.

Ans. -ax-2bc+562-y2. 10. From a2b+3abc+b2c-7, subtract a2b-abc+b2c. Ans. a2b+4abc+1b2c-7.

When one quantity is to be subtracted from the Sum of two or more quantities, the operation will be expedited by changing the signs of the subtractive quantity, and then adding all the quantities together.

11. From the Sum of 2a+3bc-1, and 3a-4bc+3, subtract 4a-bc-2+y. Ans. a+4-y. 12. From the Sum of ax2-662+3ay2, and 2ax2+6b2—ay2, subtract -3ax2+b2+2ay2-5. Ans. 6ax2-62+5.

13. From the Sum of 3ab+bc-5x2, and ab-3bc+x2-3, subtract 5bc-3x2-4+y2. Ans. 4ab-7bc-x2+1-y2.

14. From the Sum of 5x-3xy+7y2, and 5xy-3x-4y2-9, subtract -x+xy-3y2+7.

15. From the Sum of 2a-b+3cd,

cd, subtract 4a-5b-cd.

Ans. 3x+xy+6y2—16. 5a+3b-cd-3, and a+2b-.. Ans. 4a+96+2cd-3.

16. From the Sum of 3x2+y-5xy, 2x2-3y+xy, and 4y-3x2 -xy+5, subtract x2+5y-xy—3.

Ans. x2-3y-4xy+8.

17. From the Sum of 2ay2-2b+3ax, 2ay2+b-ax, and 36— ay2+3-y, subtract 3ay2+46-ax+y.

Ans. -2b+3ax+3-2y.

18. From the Sum of 3a2+xy2-2by, 5a2+3xy2-3by, and 3xy2+4a2+by, subtract 2a2-xy2—5by+5.

Ans. 10a2+8xy2+by−5. 19. From the Sum of a262-3y2+5xy, 3a2b2+3y2-2xy, and 5y2+3a2b2 —xy+6, subtract a2b2+xy-y2-3.

Ans. 6a2b2+6y2+xy+9.

20. From the Sum of 5y2-3ax-2bc, 4ax-2y2+5bc, and 3ax+y2-bc+5, subtract ax-bc+3y2-1.

Ans. y2+3ax+3bc+6.

21. From the Sum of a2y-x2+bc-d, 3x2-3bc+5d, and a2y+3x2-4bc-d, subtract 5x2 -2bc+7d.

Ans. 2a2y-4bc-4d.

22. From the Sum of 5b2+3ax+2y, 3b2—ax−y+1, and 4ax -62-5y+2, subtract 362+2ax-y+10.

Ans. 4b2+4ax-3y-7. 23. From the Sum of 7a+6x2-y2+1, x2+3y2-c-3, and 7x2+y2—3c+5, subtract 5a+2x2-2y2+5c+2. Ans. 2a+12x2+5y2-9c+1.

24. From the Sum of 2 (a−x)+y, 3(a−x)+2y, and 5(a—x)—y, subtract 4(a-x)+y-2, (32). Ans. 6(a-x)+y+2. 25. From the Sum of 26(a+c)2+3, b(a+c)2−1, and 36(a+c)2 -2, subtract the sum of b(a+c)2+b, and 5b(a+c)2 —b.

Ans. 0.

Indicated Subtraction.

(37.) It is sometimes expedient merely to indicate the subtraction of a quantity, without performing the operation.

To denote the subtraction of a positive monomial, nothing more is necessary than to place the sign before it; thus a-b denotes that b, that is, positive b, is to be subtracted from a.

The subtraction of a negative monomial, will be denoted by enclosing the quantity, with its negative sign, in a ( ), and prefixing the sign to the parenthesis.

a.

Thus a-(-b) denotes that negative b is to be subtracted from
When the subtraction is performed, the expression becomes a+b.

The subtraction of a polynomial will be denoted by enclosing the polynomial in a ( ), and prefixing the sign to the parenthesis.

Thus a-(b+c+d); in which the sign denotes that the enclosed polynomial is to be subtracted. When the subtraction is performed, the expression becomes a-b-c-d.

Change of Signs in a Polynomial.

(38.) The signs of the value of a Polynomial, will evidently be changed, by changing the signs of all the terms of the Polynomial.

The value of a Polynomial is therefore not affected by changing the signs of any or all of its terms, enclosing those terms in a ( ), and prefixing the sign -to the parenthesis.

Thus a+b-c is equivalent to a−(−b+c), or a—(c—b); for if (c-b) be subtracted, as is required by the sign prefixed to it, the result will be a+b-c.

26. Under what different forms may the value of the polynomial ab+2cd-3x+5 be expressed?

27. Under what different forms may the value of the polynomia ax-3y+4b2-5c-7 be expressed?

MULTIPLICATION.

(39.) Algebraic MULTIPLICATION consists in finding the Product of one quantity taken as many times, additively, or subtractively, as there are units in another quantity.

The quantity to be multiplied is called the multiplicand, and the multiplying quantity the multiplier: both together are called the factors of the product, (12).

When the Multiplier is positive, the multiplicand is repeated positively, or is repeatedly added.

Thus 5a ×3, 5a multiplied by positive 3, is 5a+5a+5a, equal to 15a; the multiplicand 5a being repeatedly added.

When the Multiplier is negative, the multiplicand is repeated negatively, or is repeatedly subtracted.

Thus 5a X-3, 5a multiplied by negative 3, is -5a-5a-5a, which is equal to -15a; the multiplicand 5a being repeatedly subtracted

Multiplication of Monomials.

(40.) The Product of two monomials consists of the product of their coefficients multiplied into all their literal factors.

For example, 3ac × 2x is equal to 6acx; for the factors may be taken in the order 3× 2acx, which becomes 6acx by substituting 6 for 3 x 2.

What is the Product of 3a x 46? and how do you reason in finding that product? What is the Product of 5a2bx 2x? and how do you reason in finding it? Of 7 ac2 xxy? Of xy2×5? Of 3 x 7a2b2?

(41.) When the same letter occurs in both the monomials multiplied together, its exponent in the Product will be the sum of its exponents in the two factors.

Thus 3a2x × 2ax, or 3a2x1 × 2a1x1, is equal to 6a2axx, (40), and his product becomes 6a3x2 by substituting a3 and x2 for their equiva ents a2a and xx, (13) and (16).