No change in the process is necessary when some or all of the coefficients are fractional.

Ex. 39. Find the product of 4 an+1 3 an+7 an-1 and 5 an+1 - 2 an.

The process of multiplication is not affected in any way because the letter n appears as an exponent. We shall consider for the present that n represents a positive whole number. The separate partial products are obtained by adding the exponents of the like factors entering into them. E. g. The first term of the multiplicand may be multiplied by the first term of the multiplier as follows:

4 an+1 × 5 an+1 = 20 a(n+1)+(n+1) = 20 a2n+2.

The remaining partial products may be found in a similar way.

n

42. 5x+4x31 + 3 x2n + 2 x" by 2 x − 1. 43. a"-863 a"-262 + a"-1b by a3b” — a^b3. 44. 3a+1 7 am + 4 am-1 by 2 am - 5 am-1 ̧ 45. x2n + x2 + 1 by a2n

Xn 1.

n-1,,m+1

46. x2+1yn−1 + x2-1ym+1 by x2-1ym+1 — xm+1yn−1. 47. x2n+1

x2 + x2-1 by x2n−1 + x2n-2 + x2n-8. 48. x2-a + x + x2-c by xn+a + anto + xente.

49. am++ a + 1 by a"-"

am + 1.

15. Removal of Parentheses. Parentheses may be removed by applying the Distributive Law for Multiplication.

Ex. 1. Simplify 6a-5{a - 4 [3 + 2 (a− 1)]}.

We have, 6a5{a − 4 [3 + 2 (a1)]}= 6a-5{a-4 [3+2a - 2]}

EXERCISE VII. 5

Simplify each of the following expressions :

1. 1+ 2{1 + 3 [1 + 4 (1 + 5 x)]}.

2. x{(y − z) — [x + y − z − 2 (x − y + z)]}. 3. 2+2{22 [2 + 2 (2 − 2 x)]}

4. (x + 1) − 2{(x + 2) + 3 [(x + 3) − 4 (x + 4)]}.

·

6. 5{4 [3 (2 + a)]} − 5 {− 4 [− 3 (2 − a)]}. 7. 7{b-4[b-4 (b + y)]} — 6 {b

− 4 [b − 2 (b − y)]}. c [1 + b (1 + a)]}. 9. ab c[ab (a + b + c) − (b + a)] − c − b} .

8. a{1 + b[1+ c (1 + d)]} − d{1 +

10. bb{b + c [a (b − c) + b (c − a) + c (a - b)]}·

STANDARD IDENTITIES

16. Special Products. Just as in Arithmetic we find it necessary to commit to memory the Multiplication Table, so in Algebra certain products occur so frequently that it is important to memorize them.

17. A polynomial expression is said to be an expansion of a second polynomial expression if it is obtained by raising the second expression to some power.

E. g. The expansion of (a + b)2 is a2 + 2 ab + b2.

18. Square of a Binomial Sum.

Theorem I. · (a + b)2 = a2 + 2 ab + b2.

The square of a binomial sum is equal to the square of the first term, increased by twice the product of the two, plus the square of the second.

MENTAL EXERCISE VII. 6

Expand each of the following binomial sums:

33. (7 t + 20 z)2. 34. (18k 10 n)2. 35. (19a2c)2. 36. (3b+18 d). 37. (20 c + 5 g)2. 38. (10b+ 16k)2. 39. (14 d + 5 n)2.

40. (12 h + 11 r)2.

41. (21s+ 2 v)2.
42. (8 ab + 5)2.
43. (9 xy + 10)2.
44. (11a4bc)2.
45. (7x+4yz)2.
46. (6 ab 5 cd)2.
47. (10xw+9yz)2.

48. (15 ac+5bd)2.

Theorem II. (a - b)2 = a2 – 2 ab + b2.

The square of a binomial difference is equal to the square of the first term, diminished by twice the product of the two, plus the square of the second.

Check. Let x = 6.

Ex. 3. (3x-5 y)2= (3x)2 -2.3x 5y+(5 y)2 (3x-5y)2

MENTAL EXERCISE VII. 7

Expand each of the following binomial differences :

21. (8 g − 1)2.
22. (3 k-5)2.

23. (4h -7)2.
24. (6 v-5)2.
25. (7 w-9)2.
26. (86 s)2.
27. (11 - 4 t)3.
28. (13 r -3)2.
29. (2 s — 15)2.
30. (4q8)2.
31. (9a 3b)2.
32. (6c7d)2.
33. (5h 12 r)2.

-

34. (9p7q)2.

20. Multiplication of the Sum of Two Terms by their

Difference.

Theorem III. (a+b)(a - b) = a2 — b3.

The product obtained by multiplying the sum of two terms by their difference is equal to the difference of the squares of these terms.

Let the student check each of the examples above.

35. (6c+12 g) (6 c 12 g).
36. (4814) (48-142).
37. (3 ab + 5) (3 ab-5).
38. (2 cd +9) (2 cd-9).
39. (7+6my) (7 — 6 my).
40. (8 + 9 nz) (8 — 9 nz).
41. (4 ab + 5 c)(4 ab — 5 c).
42. (10x + 7yz) (10 x − 7 yz).
(11 bk + 9 w) (11 bk — 9 w).
(8 gx + 10 hy) (8 gx

43.

— 9 s).

44.

10 hy).

46.

29. (15 q + 12 z) (15 q — 12 z). 30. (17 d + 6 v) (17 d − 6 v). 31. (18 n+38) (18 n-3 s). 32. (10p+ 16 t) (10 p-16 t). 33. (9q+20 r)(9 q- 20 r). 34. (19a5h) (19a - 5 h).

(7 abc + 8 d) (7 abc — 8 d). 47. (14a+11yzw) (14x-11yzw). 48. (15 ab+ 10cd) (15 ab- 10cd). 49. (16bc+15mn) (16bc-15mn). 50. (17mn +18pq) (17mn-18pq).

21. Square of a Polynomial. Consider the square of a polynomial consisting of three terms:

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

= (a+b+c) a

+(a+b+c) b

+(a+b+c) c.

From the arrangement of the work above it appears that each partial product obtained by multiplying any one of the terms in parentheses by the factor outside must be of the second degree. The only possible terms which can arise in this way will be those which are the squares of the given letters, such as a2, b2, and c2 and those which are the products of all possible pairs of the letters, such as ab, ac, and bc.

By examining the identity above it may be seen that a2 will occur but once, that is, as a result of the multiplication in the first line; b2 will occur but once, that is, as a result of the multiplication in the second line; c2 once only, and that from the multiplication in the third line.

Furthermore, it appears that any product such as ab, of two different letters, will occur twice and twice only.