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3. From y§a - 3x + 3b take y+a - } x.

4. Multiply c2 4c by c2-4c+.

5. Multiply x − } x2 + 1 x3 by 1 x + } x2 + 4 x3.

6. Multiply 0.5 m 0.4 m3n +1.2 m2n2+0.8 mn3 1.4 n°

by 0.4 m2

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7. Divide a1 — 7 a3b + } & a2b2 + & ab3 by a + z b.

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1. Find the value of x+y+23-3 xyz, if x = 1, y = 2,

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2. Find the value of √2bc-a, and of √2 bc-a, if b = 8, c = 9, and a = 23.

3. Add a2b ab2b3 and a3

1a2b+ ab2 — § 6o.

4. Multiply am - ambm + b2m by am+bm.

5. Multiply 4 a2m+4 + 6 am+3 + 9 a2 by 2 am+4 — 3 a3.

6. Divide +8 y3 1253 +30 xyz by x+2y-5%.

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7. Simplify (x - a)2 — (x — b)2 — (a − b) (a + b − 3 x).

8. Find the coefficient of x in the expression

x+a-2[2a-b (cx)].

9. Multiply 4 cm+2n-1 - 7 x2m-3n+2+5x2n+3m-2 by 5 x2-m-2.

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11. Divide m3y" — m1+zyl+n + m3—*yn—4 by m2*y*4.

12. Divide a1+3y .

a3 + a2

2+5y

by a2-x+21

13. Divide x

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xm−5n+4 + x2m by x2-m-2n

14. Divide y3 — y3-4m + y^p+1 by y2p-m+1.

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19. Divide 6x4m+5 13x3m+5+13x2m+5 13 xm+5

5 x5

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21. Arrange according to descending powers of x the following expression, and enclose the coefficient of each power in a parenthesis with a minus sign. before each parenthesis except the first:

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22. Divide 1.2 ax - 5.494 a3x2 + 4.8 a2x2 + 0.9 ax1

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23. Multiply a2 – } ab + b2 by fa+fb.

24. Multiply a2 + ab +

b2 by a - fb.

14

1

12

25. Divide a3 + 1 ab2 + b2 by a + b.

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26. Subtract x2 + ‡ xy + ‡ y2 from x2 - xy + } y2.

27. Subtract x2 + xy-y2 from 2x2xy + y2.

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28. If a = 8, b = 6, c = 4, find the value of

√a2 + 2 bc + √b2 + ac +√c2 + ab.

CHAPTER VI.

SPECIAL RULES.

Multiplication.

107. Square of the Sum of Two Numbers.

RULE 1.

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The square of the sum of two numbers is the

sum of their squares plus twice their product.

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RULE 2. The square of the difference of two numbers is the sum of their squares minus twice their product.

109. Product of the Sum and Difference of Two Numbers.

(a + b) (a - b) = a (a - b) + b (a - b)

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RULE 3. The product of the sum and difference of two

numbers is the difference of their squares,

110. The following rule for raising a monomial to any required power will be useful in solving examples in multiplication :

Raise the numerical coefficient to the required power, and multiply the exponent of each letter by the exponent of the required power.

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111. If we are required to multiply a+b+c by a+b−c, we may abridge the ordinary process as follows:

(a + b + c) (a + b − c) = { (a + b) + c } { (a + b) − c}

By Rule 3,

By Rule 1,

= (a + b)2 - c2

= a2 + 2 ab + b2 — c2.

If we are required to multiply a+b-c by a-b+c, we may put the expressions in the following forms, and perform the operation :

(a + b−c) (a− b + c) = {a + (b − c)} {a − (b − c)}

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112. Square of any Polynomial. If we put x for a, and y + z for b, in the identity

we have

or

(a + b)2= a2 + 2 ab + b2,

{x + (y + z) }2 = x2 + 2 x (y + z) + (y + z)2,

(x + y + 2)2

=

= x2 + 2xy + 2 xz + y2 + 2 yz + x2
= x2 + y2 + x2 + 2xy + 2 xz + 2 yz.

The complete product consists of the sum of the squares of the terms of the given expression and twice the product of each term into all the terms that follow it.

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