CASE II.

A Polynomial by a Monomial.

72.-Ex. 1. Multiply c- d+b by a.

SOLUTION. Each term of the multiplicand must be taken a times. a times c is ac; a times -d is - ad, and a times b is ab. Hence, c-d+b multiplied by a is ac-ad+ab.

2. What is the product of 2a+be by a2c?

3. What is the product of 4ac-3ax by 7ax?

c- d+ b

α

ac-ad+ab

73. Rule for Multiplying a Polynomial by a Monomial.Multiply each term of the multiplicand by the multiplier.

5. Multiply 2acx-3ay by -4ax. Ans. -Sa2cx2 + 12a2xy.

6. Multiply 3x2 - 4y2+522 by 2x2y.

7. Multiply 2xy2z3 + 3x2y3z — 5x2yz2 by 2xy2z.

8. Multiply -3+ax-1+by by -α.

Ans. 3a-a2x+a-aby.

Ans. 3x+2y-3x2+1,

Ans. Sum+1b+12amb+18abm+1.

CASE III.

A Polynomial by a Polynomial.

74.-Ex. 1. Multiply 5a-2b by a+b.

SOLUTION. (a+b) times (5a-2b) is a times (5a-26), plus b times (5a-2b). a times (5a-2b) is 5a2-2ab, and b times (5a-2b) is 5ab2b2. The sum of the partial products is 5a2+3ab - 262.

2. Multiply 2x+3y by x+y.

3. Multiply x+y by x-3y

5a - 26

a + b 5a2-2ab

5ab-2b2

5a2+3ab-2b2

75. Rule for Multiplying a Polynomial by a Polynomial.-. Multiply each term of the multiplicand by each term of the multiplier, and add the partial products.

76. The multiplication of polynomials may be indicated by enclosing each of the factors in a parenthesis and writing them one after the other; and the algebraic expression is said to be expanded when the multiplication thus indicated is performed.

1. Indicate the multiplication of 3ab2-cd by xy+9. Ans. (Sab-cd3)(xy2+9).

7. Expand (a+b+c) (a+b+c).

8. Expand (x2-x+1)(x2 +x+1)(x1 − x2+1). Ans. x3+x* + 1.

SECTION VII.

DIVISION.

77. Division is the process of finding how many times one quantity is contained in another; or,

Division is the process of separating a product into two factors, one of which is given.

The quantity to be divided or separated into two factors is called the Dividend, the quantity or factor given to divide by is called the Divisor, and the result of the division is called the Quotient.

The part of the dividend, if any, remaining undivided is called the Remainder.

EXERCISES.

78.-Ex. 1. How many times is 2a contained in 6a?

SOLUTION.--2a is contained in 6a 3 times, since the product of 3 by 2a is 6a.

2. How many times is 5x contained in 15x? In 25x?

3. How many times is a contained in aby? In 2ab?

4. What is the coefficient of the quotient of 6a divided by 2? Of 25x divided by 5x?

5. If we divide ab by a, what is the quotient? If we divide ab by b, what is the quotient?

6. If we omit from the term xy the factor x, by what will that term have been divided?

7. If we omit from the term a2 the factor a, what will be the result? If we omit from a3 the factor a, what will be the result?

8. If we divide a3 by a, what is the quotient? If we divide a by a2, what is the quotient?

9. If we divide a" by a2, the quotient is a3; what is the exponent of the factor in the quotient? What is the exponent of

the factor a in the dividend diminished by the exponent of the factor a in the divisor?

10. If we multiply +3 by +2, the product is +6; what then is the quotient of +6 divided by +3? by +2?

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11. If we multiply 3 by 2, the product is +6; what then is the quotient of +6 divided by -3? by -2?

12. What is the quotient of +10a divided by – 5a?

13. If we multiply +3 by 2, the product is -6; what then is the quotient of 6 divided by +3?

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14. What is the quotient of - 6a divided by +3a?

15. If we multiply 3 by +2, the product is -6; what is the quotient of 6 divided by -3?

16. What is the quotient of 6a divided by -3a?

79. Principles.-1. The coefficient of the quotient is equal to the coefficient of the dividend divided by the coefficient of the divisor.

Thus, 15ax+5a= 3x, for 3x × 5α = 15ax.

2. To omit a factor from a term is to divide by that factor.

Thus, axy with a omitted, or xy, is the same as axy÷a, which is equal to xy; for xy× a= = axy.

3. The exponent of a factor in the quotient is equal to its exponent in the dividend diminished by its exponent in the divisor.

Thus, a5a2, or a5 with the factor a2 omitted, is equal to a3, for

α κ α = α.

4. The quotient is positive when the dividend and divisor have like signs, and negative when the dividend and divisor have unlike signs.