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CASE II.

69. When one or both of the factors are polynomials 1. Multiply x-y+z by a+b-c.

OPERATION.

- y +z at b

Product by los

at-ay+az Product by b,

bx-by+bz Product by --C

--cx+cy-cz Entire Product,

ax-ay+az+bx-by+62-cx+cy-cz Hence the following general

RULE. Multiply all the terms of the multiplicand by each term of the multiplier, and add the partial products.

[blocks in formation]

(4.)

(5.) 3x+ 2y

20+xy-2yo 4x— by

3x-3y 12x + 8cy

6x+3x+y6xy' -15xy-10y

-6x*y3xy: +6yo 12x*— 7xy-10yo

6x23x*y–9xy' +6y 6. Multiply 3a'xc--yoz+x by 2az'.

Ans. 6a xz2ayz' +2az 7. Multiply &*_3x+2x2–5x+3 by 3x”.

Ans. 3xo_9x® +6x*—15x® +-9.3* 8. Multiply a'd- 3a'd+a'c-acta-c+1 by ac

9. Multiply 2ax-3x by 2x+4y.

Ans. 4axo+8axy—6x'—12xy. 10. Multiply 3a42ab-b' by 2–46.

Ans. 6a_16a’b+6ab?+46. 11. Multiply x-ryty' by x+y.

Ans. x+y». 12. Multiply a-3ac+c by a-c. Ans. a'_4a2c+4aco—c*. 13. Multiply 2.x*—3x+2 by x-8.

Ans. 2.'-192 +26x-16. 14. Multiply a+2al+2ab + by a'-2ab+2ab:--b'.

Ans. a--b. 15. Multiply am +3m by a" +6".

Ans. am+h ta,m tamla +bm+r. 16. Multiply 4x*+8x2 +16x+32 by 3.x-6. Ans. 12x*—192. 17. Multiply a ta’l+ab:+38 by ab.

Ans. a*-6. NOTE.—The product of two or more polynomials may be indicated, by inclosing each in a parenthesis, and writing them one after another, with or without the sign, x, between the parentheses. Such an expression is said to be expanded, when the indicated multiplication has been actually performed.

18. Expand (a+m) (a+d). Ans. a'+am+ad+dm. 19. Expand (a+2m-1) (a+1). Ans. a'+2am+2m-1. 20. Expand (+*+42° +52-24) (7_4:+11).

Ans. 2+1512—264. 21. Expand (a'_4aR+lla-24) (a®+4a+5).

Ans. a' _41a-120. 22. Expand (m-3) (m-1) (m+1) (m+3).

Ans. m-10m*+9. 23. Expand (x*—2x2+3x—4) (4x8+3x++2x+1).

Ans. 4x9–5x+8x9–10x8_8x*–5x44. 24. Expand (y +248 +yo_4y—11) (72—2y+3).

Ans. yo +10y-33. 25. Expand ( cc+1) (c+c+1) (6-C*+1).

Ans. c+0+1. 26. Expand (x8–5x+ +13x*_~*--x+2) (°—2x—2).

Ans. it'— 7.2®+21x'—17x*—25x8+6x2–2x: -4. 27 Expand (16x9—82 +4.x*—2+1) (2x+1).

Ans. 320;* +1

FORMULAS AND GENERAL PRINCIPLES.

70. A Formula is the algebraic expression of a general truth or principle.

The following formulas are useful, as furnishing rules for obtaining the products of certain binomial factors. If a and b represent any two quantities whatever, then

a+b=their sum, and

a-b=their difference; and we have, after performing the indicated operations, the results which follow: I.

(a+b)=(a+b) (a+b)=a’+2ab+89 Or, expressing the result in words,

The square of the sum of two quantities is equal to the square of the first, plus twice the product of the first and second, plus the square of the second. II.

(ab)=(a-6) (ab)=a_2ab+6 Or, in words,

The square of the difference of two quantities is equal to the square of the first, minus twice the product of the first and second, plus the square of the second. III.

(a+b) (ab=a'—6* Or, in words,

The product of the sum and difference of two quantities is equal to the difference of their squares.

By the aid of these formulas we are enabled to write the square of any binomial, or the product of the sum and difference of any two quantities, without formal multiplication.

EXAMPLES FOR PRACTICE.

1. What is the square of 3a+2ab? The

square of the first term is 9a’, twice the product of the two terms is 12a2b, and the square of the second term is 4aobo; hence, by the first formula,

(3a+2ab)=9ao+12ab+4a'b', Ans. .

2. What is the square of 2.co—5 ?

The square of the first term is 4x*, twice the product of the two terms is 20x®, and the square of the second term is 25; hence by the second formula,

(2x*–5)=4x*—20x2+25, Ans. 3. What is the product of 5x+y' and 5.—yo?

The square of 5x is 25x", and the square of ye is yo; hence by the third formula,

(5x+y) (5x-y)=25c'—-y, Ans. 4. What is the square of c+m? Ans. C+2cm+m'. 5. What is the

square
of

Ans. x*—2xy +yo. 6. What is the product of x+y and x-y? Ans. x*—yo. 7. What is the square of 3.c*+4y? Ans. 9.«*+24x*y+167'. . 8. What is the square of 50°—2cd ?

Ans. 25c-20cd+4cd". 9. What is the product of 4:+3yz and 42-3yz?

Ans. 1674_9yoz". 10. What is the square of 3aRx+2ay ?

Ans. 9a*x' +12aRxy +4a’yo. 11. What is the square of x+1?

Ans. xo +2x+1. 12. What is the square of 27°—1?

Ans. 4: -42° +1. 13. What is the product of m+1 and m-1? Ans. m-1. 14. What is the square of z—30 ? Ans. 74602 +900. 15. What is the product of 3a’b+d and 3a’b_d* ?

Ans. 9a*b*_* 16. What is the square of x-dy? Ans. **

xy + ły". 17. What is the square of 2c++?

Ans. 4c +2+. 18. What is the square of 3*+y"? Ans. 2m +23"y"+yon. 19. What is the product of x+yand x-y"? Ans. xm—yon.

71. The binomial square occurs so frequently in algebraic operations, that it is important for the student to be perfectly familiar with its form. The higher powers of any binomial may be obtained by actual multiplication. The 3d, 4th, and 5th powers, however, may sometimes be easily written, without actual multiplication, by means of the formulas which follow :

1. (a+b)=a+3aRb+3ab:+8.
2. (ab)=a_3a2b+3ab?—68.
3. (a+b)*=a*+4ab+6a*b* +4ab* +3*.
4. (ab)=a*_4a'b+6a’b?_4ab'+b*.
5. (a+b)=a'+5a*b+10aol*+10a°+5ab* +6.
6. (ab):=a_5a4b+10a%2—10a*b*+5ab-—6.
Let the pupil verify the above by actual multiplication.

72. A polynomial is said to be arranged according to the descending powers of any letter, when the terms are so placed that the exponents of this letter diminish from left to right throughout all the terms that contain it. Thus, the polynomial

X-4x+2.'-+7 is arranged according to the descending powers of x.

73. A polynomial is said to be arranged according to the ascending powers of any letter, when the terms are so placed that the exponents of this letter increase from left to right throughout the terms that contain it. Thus, the polynomial

d-ax+cx_b. is arranged according to the ascending powers of x.

74. A term or quantity is said to be independent of any letter, when it does not contain that letter. 75. The product of two polynomials has certain special properwhich

may

be stated as follows : 1.- If both polynomials are arranged according to the descending powers of the same letter, then the first term obtained in the partial products will contain a higher power of this letter than any of the other terms; and as this term can not be reduced with any of the others, it will form the first term of the entire product.

2.-If both polynomials are arranged according to the ascending powers of the same letter, then the last term obtained in the partial products will contain a higher power of this letter than any of the other terms; and as this term can not be reduced with

any

of the others, it will form the last term of the entire product.

3.- If both polynomials are homogeneous, then the product will kr. homogeneous; and the degree of any term will be expressed by the sum of the indices denoting the degrees of its two factors.

ties,

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