EXERCISE IX.

Simplify the following expressions by removing the parentheses and combining like terms.

1. (a+b)+(b+c) − (a+c). 2. (2a-b-c)-(a-2b+c).

3. (2x-y) — (2y — z) — (2 z − x).

4. (a — x − y) − (b −x+y)+(c+2y).

5. (2x−y+3z) + (− x − y − 4 z) — (3 x — 2y − z).

6. (3a−b+7c) — (2a+3b) — (5b − 4c) + (3 c − a).

7. 1 − (1 − a) + (1 − a + a2) − (1 − a + a2 — a3).

12. 2x+(y−3z) — {(3 x −2y)+z}+5x−(4 y− 3 z).

13. {(3a-2b)+(4 c − a)} - {a − (2b3a)-c}

+{a-(b-5c-a)}.

14. a- · [2a+(3 a−4a)]—5a—{6a-[(7a+8a)—9a]}. 15. 2a-(3b+2c) — [5 b − (6 c − 6 b)+5 c

16.

- {2a-(c+2b)}].

a-[2b+3c-3a-(a+b)}+{2a-(b+c)}].

17. 16 - x - [7 x — {8 x − (9 x − 3 x − 6x)}].

--

18. 2a-[3b+(2b − c) — 4c+{2a — (3b — c — 2b)}].

19. a-[2b+3c-3a-(a+b)}+2a-(b+3c)].

[5b-{a-(3c-3b)+2c (a — 2b — c)}].

56. The rules for introducing parentheses follow directly from the rules for removing them:

1. Any number of terms of an expression may be put within a parenthesis, and the sign plus placed before the whole.

2. Any number of terms of an expression may be put within a parenthesis, and the sign minus placed before the whole; provided the sign of every term within the parenthesis be changed.

It is usual to prefix to the parenthesis the sign of the first term that is to be inclosed within it.

EXERCISE X.

Express in binomials, and also in trinomials:

1. 2a-3b-4c+d+3e-2f.

2. a 2x+4y-3z-2b+c.

3. a3a-2a3-4a2+a-1.

4. —3a-2b+2c-5d—e—2ƒ.

5. ax― by — cz – bx + cy+az.

6. 2x3-3x1y+4x3 y2 — 5 x2 y3 + xy* — 2y3.

7. Express each of the above in trinomials, having the last two terms inclosed by inner parentheses.

Collect in parentheses the coefficients of x, y, z in

8. 2ax-6ay+4bz-4bx-2cx-3cy.

9. ax-bx+2ay+3y+4az-3bz-2z.

10. ax-2by+5 cz-4 bx-3 cy+az-2cx-ay+4bz. 11. 12ax+12ay+4by-12bz-15 cx+6cy +3 cz. 12. 2ax-3by-7 cz-2bx+2cx+8 cz-2cx-cy — cz.

CHAPTER III.

MULTIPLICATION OF ALGEBRAIC NUMBERS.

57. THE operation of finding the sum of 3 numbers, each equal to 5, is symbolized by the expression, 3 × 5=15, read, "three times five is equal to fifteen"; or, by the expression 5×3=15, read, " five multiplied by three is equal to fifteen."

58. With reference to this operation, this sum is called the product; one of the equal numbers is called the multiplicand; and the number which shows how many times the multiplicand is to be taken is called the multiplier.

59. The multiplier means so many times. The multiplicand can be a positive or a negative number; but the multiplier can only mean that the multiplicand is taken so many times to be added, or so many times to be subtracted.

60. If we have to multiply 867 by 98, we may put the multiplier in the form 100-2. The 100 will mean that the multiplicand is taken 100 times to be added; the -2 will mean that the multiplicand is taken twice to be subtracted.

In general, a multiplier with + before it, expressed or understood, means that the multiplicand is taken so many times to be added; and a multiplier with before it means that the multiplicand is taken so many times to be subtracted. Thus,

(1) +3 × (+5) = (+ 5) + (+ 5)+(+5), or (+ 15). (2) +3 × (−5) = (− 5) + (− 5) + (− 5), or (− 15). (3) −3 × (+5)= − (+ 5) − (+ 5) − (+ 5), or (— 15). (4) −3 × (-5)=(-5) - (-5)—(-5), or († 15).

From these four cases it follows, that, in multiplying two numbers together,

61. Like signs produce plus; unlike signs, minus.

7. (-358-417) X-79=

8. (7.512—{-2.894}) × (— 6.037+{13.963}) =

=

62. The product of more than two factors, each preceded by, will be positive or negative, according as the number of such factors is even or odd.

Thus,

-2x-3x-4= +6x-4= −24. -2x-3x-4x-5-24 x-5=+120.

9. 13 x 8x-7=

10.38 x 9x-6=

11.20.9 X-1.1x8=

12.78.3 X-0.57x+1.38 x 27.9=

13.

MULTIPLICATION OF MONOMIALS.

63. The product of numerical factors is a new number in which no trace of the original factors is found. Thus, 4× 9=36. But the product of literal factors can only be expressed by writing them one after the other. Thus, the product of a and b is expressed by ab; the product of ab and cd is expressed by abcd.

64. If we have to multiply 5a by 4b, the factors will give the same result in whatever order they are taken. Thus, 5ax-4b-5x-4x axb-20x ab=-20 ab.

65. Hence, to find the product of monomials, annex the literal factors to the product of the numerical factors.

66.

a2 X a3 =aa X aaa aaaaα = = α.

аз хаз Хая = aa X ааа X aaaa=aaaaaaaaa =

It is evident that the exponent of the product is equal to the sum of the exponents of the factors. Hence,

67. The product of two or more powers of any number is that number with an exponent equal to the sum of the exponents of the factors.