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CHAPTER VII

FACTORING

58. The Factors of any expression are the parts which, when multiplied together, will produce the expression.

It is important to note the difference between the terms of an algebraic expression and the factors of an algebraic expression. The terms of an expression are algebraically added to produce the expression, while its factors must be multiplied to produce it.

An expression is rational when none of its terms contains an indicated root, such as a square root or cube root or any other root.

Factors of rational and integral expressions only are considered in this chapter.

From the definition of the factors of an algebraic expression, it is evident that any factor of an expression is an exact divisor of that expression.

Factoring an expression usually means to break it up into its prime factors, prime having the same meaning here as in Arithmetic.

In the case of a monomial, the prime factors may be found by inspection. Thus, the factors of 10xy are, 5, 2, x, y and y.

Most polynomials may also be factored by inspection, usually by recognizing the form as being produced by some one of the special rules of multiplication.

As a knowledge of factoring is exceedingly important, the student should carefully note the characteristics of the various forms considered in this chapter.

59. Case 1. A Polynomial whose terms contain a Common Monomial Factor.-Common here means belonging to each, that is a common factor is one found in every term of the expression.

Resolve into factors: 4a2b2 6ab. By inspection, 2, a, and b, are seen to be factors of each term, therefore 2ab is said to be the common monomial factor of 4a2b2 - 6ab. Since each term contains 2ab, the expression may be divided by 2ab and the quotient obtained will be the second factor.

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Rule. When a polynomial contains a common monomial factor, determine this factor by inspection. Obtain the second factor by dividing each term of the expression by the common monomial factor.

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18. 3a4b5cd42ab5c2d4x3 + 5a3b4cd4.
19. 24m3n3 + 42a2m2n2y - 36m2n1.
20. 15t6u10+10t5u11-2012u7+10tu11.
21. 42ak + 21a2k6 14a3k728a3k8.

22. a7c2a°c3 + a5c1 a5bc5 a5c6.

60. Case 2. A Trinomial which is a Perfect Square.Referring to the rule for squaring a binomial, we see that the square of any binomial consists of the square of the first term, plus or minus twice the product of the two terms plus the square of the second term. Hence any trinomial

will be the square of a binomial, if its first and third terms are squares and the middle term is twice the product of the square roots of the first and third terms.

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Thus, 16x2 24xy + 9y2 is a perfect square, since 16x2 and 9y2 are squares and the middle term, 24xy, is twice the product of their square roots.

Hence, 16x224xy + 9y2 is the product of two equal factors, each being 4x-3y or 16x2 - 24xy + 9y2 =

(4x — 3y)2. Therefore, to factor such a perfect square, we have the following

Rule. To find one of the two equal factors, extract the square roots of the first and third terms and connect them by the sign of the middle term.

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17. 121a286a2y+169y2.

18. (a+b)2 - 4c(a + b) + 4c2.

19. (xy)2 + 6a (x − y) + 9a2.

20. 2510(cd) + (cd) 2.
20.25

61. Case 3.—A Binomial which is the Difference of Two Squares. Since (a + b) (a − b)

a2

b2 are a + b and a

=

a2 b2, the factors of

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b. Any binomial of this form

must therefore be the product of the sum and difference of the two numbers of which the two terms of the binomials

are the squares. Hence the

Rule. The difference of the squares of two numbers is the product of the sum and the difference of the two numbers.

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As 36a2x2 is the square of 6ax and 49b2y6 is the square of 7by3, the factors of 36a2x2 4962y will be the sum and the difference of the two numbers 6ax and 7by3. Therefore 36a2x2 — 49b2y6 (6ax+7by3) (6ax — 7by3).

2. Factor 25u10 16u3v8.

=

Although both terms are squares, note that the terms contain the common monomial factor us and this factor should first be separated from the binomial according to the rule of Case 1.

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As 2a

- b and 36 + d are placed in parentheses, they are treated as single numbers, like monomials, and the expression may be factored by the same rule as any binomial which is the difference of two squares. Hence

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62. Case 4. A Trinomial which is the Product of Two Binomials Having First Terms Equal.

In the preceding chapter, a rule was given for writing the product of two binomials, whose first terms are alike. Thus, (x+5) (x − 3) = x2 + 2x 15,

where the sum of the two second terms of the binomials is equal to the coefficient of the middle term and the product of the two second terms is equal to the third term of the trinomial.

Hence for any trinomial of this general form, x2 + ax + b, where a represents the coefficient of x, and b the third term of the trinomial, the rule for finding its factors is simply the reverse of the rule for finding the product.

Rule. The first term of each factor is the square root of the first term of the trinomial. The second terms of the factors are two numbers, whose product will equal the third term of the trinomial, and whose algebraic sum will equal the coefficient of the middle term.

1. Factor x2 + 11x + 28.

The first term of each factor is x.

The two second terms of the factors must be two numbers whose product is 28 and whose algebraic sum is 11. These numbers are evidently 4 and 7.

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2. Factor x2 9x + 20.

The first term of each factor is x.

The two second terms are two numbers whose product is 20 and

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