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27. Types of division. The following types of division, which may be verified by the rule just given for any particular integral value of n, should be so familiar that they may be performed by inspection.

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Give by inspection the results of the following divisions.

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CHAPTER II

FACTORING

28. Statement of the problem. The operation of division consists in finding the quotient when the dividend and divisor are given. The product of the quotient and the divisor is the dividend, and the quotient and the divisor are the factors of the dividend. Thus the process of division consists in finding a second factor of a given expression when one factor is given.

The process of factoring consists in finding all the factors of a polynomial when no one of them is given. This operation is in essence the reverse of the operation of multiplication. We shall be concerned only with those factors that have rational coefficients. 29. Monomial factors. By the distributive law, § 10,

ab + ac = a (b + c).

This affords immediately the

RULE. Write the largest monomial factor which occurs in every term outside a parenthesis which includes the algebraic sum of the remaining factors of the various terms.

Factor the following:

1. 6 a2b9c9ab6c4 - 15 aabc7.

EXERCISES

Solution: 6 a2b9c + 9 ab6c4 - 15 a4bc7 = 3 abc (2 abs + 3 b5c3 — 5 a3co).

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5. 21 abn + 6 ab2n2 - 18 a2bn2 + 15 a2b2n.

6. 10 ab2cmx 5 ab2cy + 5 ab2cz - 15 abc2m2.

7. 7 a2x3y1 — 49 ax3y1 + 14 axy3z2 — 21 a2x2y2.
8. 45 a4b2c3d 9 ab*c2d3 + 27 a3bc4d2 - 117 a2b3cd4.

30. Factoring by grouping terms. If in the expression for the distributive law, ac + be = (a + b) c,

we replace

we have

bc

c by c + d,

a (c + d) + b (c + d) = ac + ad+be+bd.

We may then factor the right-hand member as follows:

ac + ad+be+bd = a (e + d) + b (c + d) = (a + b) (c + d). This affords the

RULE. Factor out any monomial expression that is common to each term of the polynomial.

Arrange the terms of the polynomial to be factored in groups of two or more terms each, such that in each group a monomial factor may be taken outside a parenthesis which in each case contains the same expression.

Write the algebraic sum of the monomial factors that occur outside the various parentheses for one factor, and the expression inside the parentheses for the other factor.

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31. Factors of a quadratic trinomial. In this case we cannot factor by grouping terms immediately, as that method is inapplicable to a polynomial of less than four terms. We observe, however, that in the product of two binomial expressions,

(mx + n) (px + q) = mpx2 + (mq + np)x + nq,

the coefficient of x is the sum of two expressions mq and np, whose product is equal to the product of the coefficient of x2 and the last term, that is,

mq⋅ np = mp⋅ nq.

Thus, to factor the right-hand member of this equation, we may remove the parenthesis from the term in x and use the principle of grouping terms. Thus

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RULE. Write the trinomial in order of descending powers of x (or the letter in which the expression is quadratic).

Multiply the coefficient of x2 by the term not involving x, and find two factors of this product whose algebraic sum is the coefficient of x.

Replace the coefficient of x by this sum and factor by grouping terms.

Factoring a perfect square is evidently a special case under this method. Thus factor x2 + 6x + 9.

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One will usually recognize when a trinomial is a perfect square, in which case the factors may be written down by inspection.

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The factors of 1120 must be factors of 28 and 40. We seek two factors of 1120, one of which exceeds the other by 3. We note that since 40 exceeds 28 by more than 3, one factor must be greater and the other less than 28 and 40 respectively.

Since

we try

=

40,

4.7 28 and 5.8
=
5.735 and 4.8 = 32,

which are the required factors of 1120.

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