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32. Factoring the difference of squares. Under the method of the preceding paragraph we may factor the difference of squares. Thus to factor x2 - b2 we observe that the product of the coefficient of x2 and the constant term is

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RULE. Extract the square root of each term.

The sum of these square roots is one factor, and their difference is the other.

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9 a2x9y1 — 16 b3c2 = (3 ax3y2 + 4 b1c) (3 ax3y2 — 4 b4c).

33. Reduction to the difference of squares. The preceding method may be used when the expression to be factored becomes a perfect square by the addition of the square of some expression.

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34. Replacing a parenthesis by a letter. Any of the preceding methods may be applied when a polynomial appears in place of a letter in the expression to be factored. It is frequently desirable for simplicity to replace such a polynomial by a letter, and in the final result to restore the polynomial.

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In this example the factor (a - b) might have been replaced by a letter.

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6. (xy) (3 a + 4b) (4 a 5b) (x − y) - (x − y) (2a-8b).

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35. Factoring binomials of the form a"b".

By § 27,
an - bn =
(a − b) (a”-1 + an−2b + an−3b2 + + br−1).
a" + b" = (a + b) (a" -1 — a"−2b + a" −3b2 — · · · + b” −1),

where n is odd.

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One can factor by inspection any binomial of the given form by reference to these equations.

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36. Highest common factor. An expression that is not further divisible into factors with rational coefficients is called prime.

If two polynomials have the same expression as a factor, this expression is said to be their common factor.

The product of the common prime factors of two polynomials is called their highest common factor, or H.C.F.

The same common prime factor may occur more than once. Thus (x − 1)2(x+1) and (x − 1)2 (x − 2)2 have (x − 1)2 as their H.C.F.

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37. H.C.F. of two polynomials. The process of finding the H.C.F. is performed as follows:

RULE. Factor the polynomials. The product of the common prime factors is their H.C.F.

EXERCISES

Find the H.C.F. of the following:

1. 4 ab2x4 - 8 ab2x2 + 4 ab2 and 6 abx2 + 12 abx +6 ab.

Solution :

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38. Euclid's method of finding the H.C.F. When one is unable to factor the polynomials whose H.C.F. is sought, the problem may nevertheless be solved by use of a method which in essence dates from Euclid (300 в.c.). The validity of this process depends on the following

PRINCIPLE. If a polynomial has a certain factor, any multiple of it has the same factor.

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K and a, b,

be represented by F and G respectively. The letters A, B, ......., l represent integers, and m, the degree of G, is no greater than n, the degree of F. We seek a method of finding the H.C.F. of F and G if any exists. Call Q the quotient obtained by dividing F by G, and call R the remainder. Then (§ 26)

F=Q.G+R,

(1)

where the degree of R in x is not so great as that of G. Now whatever the H.C.F. of F and G may be, it must also be the H.C.F. of G and R. For since F - QG = R,

the H.C.F. of F and G must be a factor of the left-hand member, and hence a factor of R, which is equal to that member. Also every factor common to G and R must be contained in F, for any factor of G and R is a factor of the right-hand member of (1), and hence of F.

Thus our problem is reduced to finding the H. C. F. of G and R. Let Q1 and R1 be respectively the quotient and remainder obtained in dividing G by R.

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where the degree of R1 in x is not as great as that of R. By reasoning similar to that just employed we see that the H.C.F. of G and R is also the H.C.F. of R and R1. Continue this process of division.

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RkQk+2Rk +1 + Rk +3,

either R is exactly divisible by Rz+1 (i.e. Rk +3 = 0), or Rx +3 does not contain x. This alternative must arise since the degrees in x of the successive remainders R, R1, R2, · are continually diminishing, and hence either the remainder must finally vanish or cease to contain x. Suppose Rx+3=0. Then the H.C.F. of R and R +1 is R+1 itself, which must, by the reasoning given above, be also the H.C.F. of F and G. If Rx + 3 does not contain x, then the H.C.F. of F and G, which must also be a factor of Rk +3, can contain no x, and must therefore be a constant.

Thus F and G have no common factor involving x.

This process is valid if the coefficients of F and G are rational expressions in any letters other than x.

39. Method of finding the H.C.F. of two polynomials. The above discussion we may express in the following

RULE. Divide the polynomial of higher degree (if the degrees of the polynomials are unequal) by the other, and if there is a remainder, divide the divisor by it; if there is a remainder in this process, divide the previous remainder by it, and so on until either there is no remainder or it does not contain the letter of arrangement. If there is no remainder in the last division, the last divisor is the H.C.F. If the last remainder does not contain the letter of arrangement, then the polynomials have no common factor involving that letter.

In the application of this rule any divisor or remainder may be multiplied or divided by any expression not involving the letter of arrangement without affecting the H.C.F.

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