CHAPTER VI

SPECIAL RULES FOR MULTIPLICATION AND

DIVISION

47. If we multiply a + b by a + b, that is, square a + b,

a + b

a + b

a2 + ab

+ ab + b2 a2+2ab+b2

the product is seen to consist of three terms, the first term being the square of a, the first term of the binomial a + b, the second term of the product, 2ab, is twice the product of the two terms of the binomial a + b, and the third term of the product, b2, is the square of the second term of the binomial a + b.

It is evident, that if, instead of a + b, the binomial had been x + y, the form of the product would have been the same, the letters changing to x and y instead of a and b. Experience then, teaches us that the square of the sum of two numbers is always equal to the square of the first number plus twice the product of the first by the second number, plus the square of the second number. Hence it is no longer necessary to multiply a binomial of this form by itself to find its square, but simply to write the product according to the rule found above. In the same way, we can find rules for writing the products of many other forms of multiplications or the quotients of certain forms of division. This so-called Inspection Method of writing the product or quotient according to rule, is of great importance in Algebra, as most of our work is done in this way. The student should take great pains to acquire the

ability of recognizing the forms of algebraic expressions and memorize the rules that apply to them.

48. The Square of the Sum of Two Numbers. Since, as shown above,

(a + b)2 = a2 + 2ab + b2

we have the following

Rule 1. The square of the sum of two numbers is equal to the sum of the squares of the numbers plus twice their product.

the product differs from the square of a + b, in the sign of the middle term, -2ab.

Hence, the following

Rule 2.-The square of the difference of two numbers is equal to the sum of the squares of the numbers minus twice their product.

50. The Product of the Sum and Difference of Two Numbers.

Here the middle term disappears and our product consists of the square of the first number minus the square of the second number, hence,

Rule 3. The product of the sum and difference of two numbers is equal to the difference of their squares.

5. (2x2 — y2)(2x2 + y2). 10. (5u2v2 + 1)(5u2v2 — 1).

51. The Product of Two Binomials whose First Terms are Alike.

Examining this product, we get the following

Rule 4. The product of two binomials, whose first terms are alike, consists of three terms; the first is the square of the first term of the binomials, the second term of the product

is the first term of the binomials with a coefficient equal to the algebraic sum of the two second terms of the binomials, and the third term of the product is the product of the two second terms of the binomials.

Multiply xa by x-b.

As the second terms of these binomials cannot be combined by addition, like arithmetical numbers, the algebraic addition must be indicated.

Similarly, the rule must apply even if the terms of the binomials are compound expressions, consisting of more than one term. Multiply [x + 3(y + z) } [x − 5(y + z)]

20. [(a + b) — 4(c + d) ][(a + b) — 3(c + d)].

52. The Product of Any Two Binomials.

Multiply 2a+3b by 3a 4b.

In this case the product is not written according to any special rule. The multiplication is performed in the usual way, except, that the numbers are not written under each

other as in an ordinary multiplication and that the partial products, if like terms, are added mentally.

Here, the partial products, as indicated by lines, 9ab and -8ab, being like terms, are added mentally, giving ab as the middle term. If the partial products are unlike terms, they are written as they are produced by multiplication.

53. Square of Any Polynomial.-If any polynomial, such as xyz, is multiplied by itself, the product is, x2+2xy - 2xz + y2 - 2yz + z2.

This product consists of the sum of the squares of the several terms and the algebraic sum of the double products of each term by each of the others.

Hence, to square any polynomial, we have the following Rule 5.-The square of a polynomial is equal to the sum of the squares of the several terms, plus twice the product of each term by each of the following terms.