-

2. Add together 3 a2b−5 ab2+7 b3 and 2 a3 — § a2b+5 ab2 — § b3. 3. Multiply ax2 - x + a by ax2 + x + a, and find the square of

2a+b-3c.

4. Simplify (x+y)2 — (x+y) (x − y) − {x (2y-x)—y (2x-y)}.

5. The product of two algebraical expressions is

x2 + x5y + x+y2 — x3y3 + y®,

and one of them is x2 + xy + y2; what is the other?

6. Prove

(i.) a(b −c)+ b(c − a) + c (a — b) = 0,

(ii.) a2(b-c)+b2 (c-a)+c2(a−b)+(b−c) (c− a) (a−b) = 0.

D. 1. If a = 1, b=2, and c=-3, find the value of 6 a+3b+4c, and of a3 + b3 + c3 - 3 abc.

2. Simplify 3 {x − 2(y − z)} − [4 y + {2 y − ( z − x)}].

3. What must be added to (a + b + c)2 that the sum may be (a − b −c)2?

4. Multiply a3 + b3 by a

b, and divide the result by a + b.

5. Divide x3 + 23 by x + z, and from the result write down the quotient when (x + y)3 + z3 is divided by x + y + 2.

6. Prove

(i.) (x + y)(x − y) + (y + z) (y − 2) + (z + x) (z − x) = 0, (ii.) (xy)2+(y − z)2 + (z - x)2

=

= 2(x − y) (x − z) + 2(y − z) (y − x) + 2( z − x)(z − y).

E. 1. Find the value of

2 √(a2 + b2) + † (a2 + b2 − c2 + 6 d2)

2. Simplify 4x − (y − x) − 3 {2 y − 3(x + y)} and

4(x −- 2 y) + 5 {3 x − 2 (x − y)},

and multiply the two results together.

3. Show that (a + b)3 = a3 + b3 + 3 ab(a + b), and verify the result when a = 1 and b =

-=

2.

4. Find the coefficient of x2 in the product of x2 - (a - b)x — ab and x2 + (a + b)x + ab.

5. Divide the difference of (2a + 3b)2 and (3a+2b)2 by a+b; divide also the sum of (2 a − 3b)3 and (3a - 2b)3 by a – b. 6. Show that

(a+b+c-d) (a + b − c + d) + ( a − b + c + d) ( − a + b + c + d) = 4(ab + cd).

F. 1. Find the value of

(a + b) (c + d) − (b + c) (d + a)

=

2. Find the value of (a - x) (a − 1 y) (a − } z).

3. If x2+7x+c is exactly divisible by x+4, what is the value of c?

4. Divide x2 — y2 — z2 + 2 yz by x + y -2, and find the coefficient of x in the quotient obtained by dividing 8x4 + xy3 — y‡ by x-y.

5. Show that

(a2+ b2) (c2+ d2) = (ac + bd)2+(ad — bc)2= (ac — bd)2+ (ad + bc)2. 6. Show that

(b − c)2 + (a − b) (a — c) = (c − a)2 + (b − c) (b − a)

G. 1. Simplify 5 x − 3[2x + 9 y − 2 {3 x − 4 (y — x)}].

2. Subtract 6 a2 - 4 ab +7b2 from 2 a2 4 ab+2b2, and from the remainder subtract 4 a2 – 6 ab + 5 b2.

3. Multiply a2 + b2 + 1 − ab — a − b by a + b + 1.

4. Divide a3 + b3 − 8 c3 + 3 a2b + 3 ab2 by a + b −2 c.

5. The product of two algebraical expressions is x 64 x, one of the expressions is x2 - 2x+4; what is the other?

6. Prove that

and

(1 + x + x2) (1 − x + x2) (1 − x2 + x1) (1 − x1 + x8) = 1 + x8 + x16, and that

a3 (b-c)+b3(c-a)+c3 (a−b) + (b−c) (c− a) (a−b) (a+b+c) = 0.

CHAPTER VI.

SIMPLE EQUATIONS.

90. A statement of the equality of two algebraical expressions is called an equation, and the two equal expressions are called the members or sides of the equation.

When the statement of equality is true for all values of the letters involved, the equation is sometimes called an identical equation. An identical equation is, however, generally called an identity, and the name equation is reserved for those cases in which the equality is only true for certain particular values of the letters involved.

=

Thus a + a 2 a, and (a + b)2 = a2 + 2 ab + b2, which are true for all values of a and b, are identities; and 5 a + 2 = 12, which is only true when a is 2, is an equation.

In an identity the sign is frequently used instead of the sign =. Thus, a + a = 2 α.

NOTE. For the sake of distinction, a quantity which is supposed to be known, but which is not expressed by any particular arithmetical number, is represented by one of the first letters of the alphabet, a, b, c, etc., and a quantity which is unknown, and which is to be found, is represented by one of the last letters of the alphabet, x, y, or z.

91. To solve an equation is to find the value, or values, of the unknown quantity for which the equation is true; and these values of the unknown quantity are said to

satisfy the equation, and are called the roots of the equation.

Def. In more precise terms we therefore define a root of an equation to be any quantity which, when substituted for the unknown quantity, reduces the equation to an identity.

This definition supplies us with a convenient test for a solution. Thus 2 is a root of

because when 2 is substituted for x, the equation is reduced to the identity

8-8+2= 2.

Ex. 1. Show that 1 is a root of each of the following equations:

Ex. 2. Show that a is a root of each of the following equations:

92. An equation which contains only one unknown quantity, a suppose, is said to be of the first degree when x occurs only in the first power; it is said to be of the second degree when a2 is the highest power of a which occurs, and so on.

Equations of the first degree are, however, generally called simple equations, and equations of the second degree are generally called quadratic equations.