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RULE. Write the product of the numerical coefficients, followed by all the letters that occur in the multiplier and multiplicand, cach having as its exponent the sum of the exponents of that letter in the multiplier and multiplicand.

EXAMPLE. 4a2b10cd4. (- 16 aa1bd7) = — 64 a6b11cd11.

17. Multiplication of monomials by polynomials. By the distributive law, § 10, we can immediately formulate the RULE. Multiply each term of the polynomial by the monomial and write the resulting terms in succession.

EXAMPLE.

9a2b2-2 ab + 4 ab2.

a + be
3 a2b

27 a4b3 - 6 a3b2 + 12 a3b3 – 3 a3b + 3 a2b7

18. Multiplication of polynomials. If in the expression for the

distributive law, a (c + d) = ac + ad,

we replace a by a + b, we have

which affords the

(a + b) (c + d) = ac + bc + ad + bd,

RULE. Multiply the multiplicand by each term of the multiplier in turn, and write the partial products in succession.

To test the accuracy of the result assume some convenient numerical value for each letter, and find the corresponding numerical value of multiplier, multiplicand, and product. The latter should be the product of the two former.

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2 a3

- 4 b3 — b2

Check : Let ab 1

=8

=1

+ 2 a3b + a2b2 + 4 ab3

a2b+ 4 ab2 + ab − 4 b3 — b2 + 2 a3b + a2b2 + 4 ab3 = 8

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(k) xa+b

+ x2a + x2b+3a-b and xa-b 1.

(1) 15x+ 11x3+6x2+2x-1 and — 3x2 - 1.

(m) 4x4 - 8 xy3 + } x2y2 — § x2y — x y and 42 xy.

2. Expand (x + y)1.

3. Expand and simplify

(x2 + y2 + z2)2 − (x + y + z) (x + y − z) (x + z − y) (y + z − x).

19. Types of multiplication. The following types of multiplication should be so familiar as merely to require inspection of the factors in order to write the product.

RULE. The product of the sum and difference of two terms is equal to the square of the terms with like signs minus the square of the terms which have unlike signs.

EXAMPLES.

(a - b) (a + b) = a2 — b2.

(4x2-3 y2) (4x2 + 3y2) = 16 x 9 y1.

20. The square of a binomial. This process is performed as follows:

RULE. The square of a binomial, or expression in two terms, is equal to the sum of the squares of the two terms plus twice their product.

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21. The square of a polynomial. This process is performed as follows:

RULE. The square of any polynomial is equal to the sum of the squares of the terms plus twice the product of each term by each term that follows it in the polynomial.

EXAMPLE. (a + b + c)2 = a2 + b2 + c2 + 2 ab + 2 ac + 2 bc.

22. The cube of a binomial. This process is performed as follows:

RULE. The cube of any binomial is given by the following expression: (a + b)3 = a3 + 3 a2b + 3 ab2 + b3.

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23. Division. By the definition of division in § 6, we have

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where n and m are positive integers and n > m.

If n = m, we preserve the same principle and write

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(I)

24. Division of monomials. Keeping in mind the rule of signs for division given in § 6, we have the following

RULE. Divide the numerical coefficient of the dividend by that of the divisor for the numerical coefficient of the quotient, keeping in mind the rule of signs for division.

Write the literal part of the dividend over that of the divisor in the form of a fraction, and perform on each pair of letters occurring in both numerator and denominator the process of division as defined by equation (I) in the preceding paragraph.

EXAMPLE. Divide 12 a4b11c2d by - 6 a2bc2d8.

-

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25. Division of a polynomial by a monomial. This process is performed as follows:

RULE. Divide each term of the polynomial by the monomial and write the partial quotients in succession.

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26. Division of a polynomial by a polynomial. This process is performed as follows:

RULE. Arrange both dividend and divisor in descending powers of some common letter (called the letter of arrangement). Divide the first term of the dividend by the first term of the divisor for the first term of the quotient.

Multiply the divisor by this first term of the quotient and subtract the product from the dividend.

Divide the first term of this remainder by the first term of the divisor for the second term of the quotient, and proceed as before until the remainder vanishes or is of lower degree in the letter of arrangement than the divisor.

When the last remainder vanishes the dividend is exactly divisible by the divisor. This fact may be expressed as follows:

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When the last remainder does not vanish we may express the result of division thus:

dividend
divisor

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remainder.

divisor

The coefficients in the quotient will be rational numbers if those in both dividend and divisor are rational.

EXERCISES

Divide and check the following:

1. 8a36a2b + 9 ab2 + 9 b3 by 4 a + b.

Solution : 4 a + b ]8 a3 + 6 a2b + 9 ab2 + 9b3| 2 a2 + ab + 2 b2

8 a3 + 2 a2b

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20. 3x36x2y + 9 x2 + 2 xy2 + 5 y3 + 2y + 9 x2 + 6 y2 + 3 by x + 2y + 3.

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