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Multiplication of Polynomials.
1. Multiply x2 + y2 by x+y.
Ans. 23 + x2 y + xy2 + y2. 2. Multiply x5 + xy6 + 7 a x by a x + 5 a x.
Ans. 6 a x6 + 6 a x2 y5 + 42 a2 x2. 3. Multiply - a by b.
- ab. 4. Multiply a by — b.
Ans. - ab. 5. Multiply - a by - b.
Ans. ab. 6. Multiply - 3 a by 14 c.
42 a c. 7. Multiply -6 a3 b2 by - 11 a 63 c.
Ans. 66 a4 65c. 8. Find the continued product of a,
–a, –a, and
Ans. a4. 9. Find the continued product of -a2b, cae, -a, -e3x2, C, – 2 a 3, 3 ab elor, — 7, and 63,23.
Ans. 42 a5 65 c3 e5 27. 10. Find the continued product of 7 abx,
X, 62 x7, -26,-3, and — 5 a7 63 25.
Ans. A_ 210 a9 67 215. 11. Multiply a + b by c+d.
Ans. ac+ad+bc+bd. 12. Multiply a3 + b2 − c by a2 - 63.
Ans. a5 - a3 b3 + a2b2 - ac-65 +63c. 13. Multiply a +b+c by a+b
Ans. a2 + 2 ab +62 — c2. 14. Multiply 22 - 32 -7 by 1 - 2.
Ans. 23-5x2 -- +14. 15. Multiply a2 +at+a6 by a2 — 1. Ans. a8 - a2.
16. Multiply 8 ao b3 + 36 a8 b4 + 54 a7 65 +27 26 66 by 8 ao 13 — 36 a8 64 +54 a7 65 – 27 a6 66.
Ans. 64 a18 66 - 432 a1668 +972 al4 610 - 729 a12 312.
Product of Sum and Difference; of Homogeneous Quantities.
17. Find the continued product of 3x + 2 y, 2x-3y, & + y, and — 2 x — y. Ans. 12 z* — 16237 — 13 22 y2 +11 2y3 +6 ye.
. 18. Multiply a + b by a - b.
Ans. 42 - 62. 19. Multiply 2 a3r+7a2 25 by 2 a3r - 7 aa 25.
Ans. 4 a6 22 - 49 a4 x10.
27. Corollary. The continued product of several monomials is, as in examples 8 and 9, positive, when the number of negative factors is even; and it is negative, as in example 10, when the number of negative factors is odd.
28. Corollary. The product of the sum of two numbers by their difference is, as in examples 18 and 19, equal to the difference of their squares.
29. Theorem. The product of homogeneous polynomials is also homogeneous, and the degree of the product is equal to the sum of the degrees of the factors.
Demonstration. For the number of factors in each term of the product is equal to the sum of the numbers of factors in all the terms from which it is obtained; and, therefore, by art. 15, the degree of each term of the product is equal to the sum of the degrees of the factors. Thus, in example 16, the degree of each factor is 12, and that of the product is 12 + 12 or 24.
Division of Monomials.
30. Problem. To divide one monomial by another.
Solution. Since the dividend is the product of the divisor and quotient, the quotient must be obtained by suppressing in the dividend all the factors of the divisor which are contained in the dividend, and simply indicating the division with regard to the remaining factors of the divisor. Hence, from art. 25,
Suppress the greatest common factor of the numerical coefficients.
Retain each letter of the divisor or dividend in the term in which it has the greatest exponent, and suppress it in the other term ; and give it an exponent equal to the difference of its exponents in the two terms. But when a letter occurs in only one term, it is to be retained in that term, with its exponent unchanged.
The required quotient is, then, equal to the quotient of the remaining portion of the dividend divided by that of the divisor, and may be indicated as in art. 8; or, when the divisor is reduced to unity, the quotient is simply equal to the remaining portion of the dividend.
The sign of the quotient must, from art. 26, be the same as that of the divisor when the dividend is positive, and it must be the reverse of that of the divisor when the dividend is negative; whence we readily obtain the rule.
Division of Monomials.
When the divisor and dividend are both affected by the same sign, the quotient is positive; but when they are affected by contrary signs, the quotient is negative.
The rule for the signs in both division and multiplication may be expressed still more concisely as follows.
Like signs give +; unlike signs give
13 b 1. Divide 65 a b by 5 a.
1 2. Divide 132 a5 63c by 11 a3 63. Ans. 12 a2c. 3. Divide 144 a b3 c2 dee by — 112 ab4ce7h.
7 be6 h
27 4. Divide 135 by - 5 a.
1 5. Divide 7 a3 x2 by 21 a5 x2.
3 a2 6. Divide am by an.
3 am Pyn - 9 7. Divide 3 am br by 4 ap 69 cm. Ans.
407 8. Divide a by
Ans. — 1. 9. Divide a by a.
Ans. - 1. 10. Divide a by
31. Corollary. If the rule for the exponents is applied to the case in which the exponent of a letter in the dividend is equal to its exponent in the divisor, when, for instance, am is to be divided by am, the exponent of the letter in the quo
Exponent equal to Zero. Negative Exponents.
tient becomes zero. But the quotient of a quantity divided by itself is unity.
Whence any quantity with an exponent equal to zero is unity.
Thus, am cam= Q =.
32. Corollary. When, in example 6 of art. 30, the exponent n of a in the divisor is greater than its exponent m in the dividend, the exponent m—n in the quotient is negative; and a negative exponent is thus substituted for the usual fractional form of the quotient. Thus, if m is zero, we have
I a : an 1: an
an In the same way we should have ab2 c3 = 2562c8d=a1b2c3-a5b2c8d1=a-4c-5d-1.
Any quotient of monomials may thus be expressed by means of negative exponents without using fractional forms.
1. Divide 5 a4 13 c d by 15 a 65 (2 d3 e.
Ans. 3-1436-%d-le-2 2. Divide 6 a76 by 9 a 87.
Ans. a6b-6=2.3-1466-6. 3. Divide 1 by 8 all b.
Ans. Ja-16-1=8-1a-116-1. 4. Divide 3 by a.
33. Corollary. Quantities, thus expressed by means of fractional exponents, may be used in all calcula