ORAL AND WRITTEN REVIEW

Exercise 14

Give orally the results of the following:

1. 5-7+11 - 8.

2. 9+15-30 +2. 3.9+5+12-17.

4. 15-24-8 +34.

5. 12-31-19.

16+17-11-34.

13.

7. 17-10+19-24.

8. 16+11+ 19 – 14.

9. 6a-5a+8a-12a+a. 10. 6a+13 a-a+11a-9 a. 11. 4 x 10+3x-5x+19. 12. 5-16+3x-8x+29.

- 8 am + 12 am - 16 am - am.

14. 12 ac 5 ac+11 ac 21 ac.

15.

-

--

4c+11-6c-18+7 c.

16. 18 x 14+ 19x+3-20 x.

17. (-4)+(-3)—(−1).

25. (-2)(-3) + (− 3) (4).

18. (-8)-(-5)+(-3). 26. (−2)(-5)–(3) (− 2). 19. (-2a)+(-3 a)-(7a). 27. —(3)(-7)+(− 2)(— 10). 20. (3a)-(-2 a) — (a). 28.

—(2) (3) (3)(−4). 21. 4a-(-5a)+(3 a−1). 29. a (2)—a (3) + a(5).

22. 5x+(-2x)-(-x+1). 30. x(-3)-x(-1) + x (− 2). 23. -3x-(-5x)−(x+2). 31. (−3)(2)(x) — (x) (− 2).

51. Simplify 2 a-[3a-2{a-3 (a−1)-2} +5a].

52. Subtract (x − a)(c — y) from (a — x)(c — y).

53 Simplify (a− c + x)2 + (c − x + α)2 + (x − a + c)2.

54. Collect (a + 1) x + (a− 1) y + (a + 2) z + (a − 1) x + (a + 1)y +(a2) z.

55. Given a minuend, 5x3 + 3x2-4x+2, and a subtrahend, 2x3+5 x2-4 x 3, find the difference.

-

56. Given a multiplicand, a3 + 4 a2 — a + 3, and a multiplier, a2 + 2 a 1, find the product.

57. Given a dividend, aa — 4 ab3 + ba

divisor, a2 + b2 - 2 ab, find the quotient.

4 a3b + 6 a2b2, and a

58. Given a product, 24-423 + 6 x2 - 4 x + 1, and a multiplicand, 12 x + x2, find the multiplier.

GENERAL REVIEW

Exercise 15

1. Show that (x − 1)2 — (x − 2) = 1 + (x −1)(x − 2).

2. Simplify (a + 1)2 — (a + 1)(a − 1) — [a (2− a) — (2 a−1)]. 3. Prove that (a + m)(a − m)+(m + 1)(m − 1)

+ (1 − a)(1 + a) = 0. 4. Divide 46 - 2 x — x3 — 2 x - 1 by 2x3- x2 — x — 1.

+

5. Add the quotient of (231) ÷ (x 1) to that of (1⁄23 − 2 x + 1) ÷ (x − 1).

6. What is the coefficient of ac in the simplified form of (ac+3)2-3 ac (ac - 1)?

7. Simplify (m+1)(m +3) (m +5) — (m−1)(m − 3) (m—5). 8. Show that (x + y + z−1)(x+y-z+1)-4(xy + z)

+(x−y+z+1)(1 − x + y + z) = 0. 9. Divide 2x2-3 xy-5 xz-2y2-5 yz-322 by 2x + y + z. 10. A certain product is 6 a1 + 4 a3y — 9 a2y2 - 3 ay3 + 2 y1, and the multiplier 2 a2 + 2 ay y2. Find the multiplicand. 11. Simplify 3[x - 2{x - 3(2x-3x+7)}].

18. Show that (1-3 x + x2)2 +x (1 − x) (2 − x) (3 − x) — 1 = 0.

13. Simplify 2x2 - 3 − (3 x + 3 x2) −x (x2 −3) − (x + 1)(2 — x2o) and subtract the result from 5 – 2 x.

14. Simplify (a + 4) (a + 3) (a + 2) − (a + 3) (a + 2) (a+1)

-(a+2)(a+1)(a + 1).

15. Prove that (1 + c2) (1 + a)2 − 2 (1 — ac) (a — c)

=(1+c)2(1 + a2). 16. Subtract a +3 from the square of a +2, and multiply the result by the quotient obtained when a3-1 is divided by a+as+a2+a+1.

17 Simplify a -- [3 a−(x−a)]+[(2 x − 3 a) — (x−2 a)]. 18. Divide c3-3 cd+d+1 by c-cd-c+d2 - d+1. 19. Find the continued product of a2 — ab+b2, a2 + ab + b2, and a1 — a2b2 + bʻ.

20. Simplify (ax3- a2x2+ax-1) (ax+1)

- (a2x2+1) (ax+1) (ux − 1). 21. Multiply 8 a3 - 27 by a +2 and divide the product by 2a-3.

22. Divide c+36 c5-18 c2- 73 c+12 by -5c+4-6 c. 23. Simplify [(22 + 3 x + 2) (x2 − 9)] ÷ [(x+3) (x2 − x − 6)]. 24. Show that x3 + y3 +1-3 xy − (1 − xy) — y(y3 — x) —

x(x2- y) = 0.

25. Divide 82 m*n* + 40 - 67 m2n3 + 18 m3n3 - 45 m3n by 3 m2n1 — 4 m2n2 + 5.

26. What must be the value of m in order that x2 + 18 x +m may be exactly divisible by x+4?

27. Show that 2(4+x2+a2-ax-2a-2x) = (2− x)2 + (a−2)2 + (x − a)2.

28. What must be the value of m+n in order that x +32-3 +2x2+mx + n may be exactly divisible by x2 + 2x+1? 29. By how much does (a2x2+3 ax + 2)2 exceed 2(3 a3 + 2 a2x2+6 ax)?

CHAPTER VI

THE LINEAR EQUATION. THE PROBLEM

87. An equation is a statement that two numbers or two expressions are equal.

88. The expression at the left of the sign of equality is the left member (or first member), and the expression at the right, the right member (or second member) of the equation.

89. An equation is a conditional equation if its members are equal for particular values of the unknown quantity.

Thus,

3x+5=x+7 is an equation only when the value of x is 1.

90. An equation is an identical equation when its members are equal for any and all values of the unknown quantity. Thus, x2−1=(x+1)(x−1) is an equation for any value of x whatsoever. A conditional equation is usually referred to as an equation; an identical equation, as an identity.

91. To solve an equation is to obtain the value of the unknown number that will, when substituted for that unknown number, make the members of the equation equal.

92. The value found to make the members of an equation equal, or to satisfy the equation, is a root of the equation. A root of an equation when substituted for the unknown quantity reduces the original equation to an identity.

93. A linear or simple equation is an equation which, when reduced to its simplest form, has no power of the unknown quantity higher than the first power. Thus:

5x15 is a linear or simple equation in x.

7y = 35 is a linear or simple equation in y.

3x + 2 = 2 x + 7 is a linear equation in x, but is not reduced in form. (x + 5)2 = x2 + 7 x + 6 is a linear equation in x, for, when simplified, the resulting equation will have only the first power of x.

While the final letters, x, y, and z, are most commonly used for representing unknown quantities in equations, any other letters may and will be used in later practice.

94. The solution of equations is based upon the truths known as

AXIOMS

1. If equals are added to equals, the sums are equal.

2. If equals are subtracted from equals, the remainders are equai.

3. If equals are multiplied by equals, the products are equal.

4. If equals are divided by equals, the quotients are equal.

5. If two quantities are equal to the same quantity, they are equal to each other.

In general, these axioms may be illustrated as follows. Given the equation A= B.

95. The same number may be added to, or subtracted from, both members of an equation.