138. Euclidean method. In case the highest common factor is not readily found by inspection of factors, a longer method, analogous to one suggested by Euclid (B.c. 300) for finding the greatest common divisor, may be employed.

139. This method depends upon two theorems :

1. A factor of an algebraic expression is a factor of any multiple of that expression.

Proof. 1. Let a, b, p, q be algebraic expressions, p and q being the factors of b.

4. I.e., if p is a factor of b, it is a factor of any multiple of b, as ab.

A similar proposition is readily seen to be true for numbers. E.g., 5 is a factor of 35; and since multiplying 35 by any integral number does not take out this 5, therefore, 5 is a factor of any multiple of 35.

2. A factor of each of two algebraic expressions is a factor of the sum and of the difference of any multiples of those expressions.

4. I.e., if p is a factor of b and b', as in step 1, then it is also a factor of the sum and of the difference of any multiples of b and b', as ab and a'b'.

A similar proposition is true for numbers. E.g., 5 is a factor of 60 and of 35, and also of the sum and of the difference of any multiples of these numbers.

140. The Euclidean method will best be understood by considering an example.

Required the highest common factor of

xx3 + 2x2-x+1 and x2 + x3 + 2x2 + x + 1.

x1 − x3 + 2 x2 −x +1\x2+ x3+2x2+ x+1[1

EXPLANATION. 1. The h.c.f. of the two expressions is also a factor

of 2 x3 + 2x, by th. 2 (§ 139).

2. It cannot contain 2x, because that is not common to the two expressions.

3. .. 2x may be rejected, and the h.c.f. must be a factor of x2 + 1. 4. x2+1 is a factor of x4 x3 + 2x2 - x + 1, by trial.

141. In order to avoid numerical fractions in the divisions, it is frequently necessary to introduce numerical factors. These evidently do not affect the degree of the highest common factor.

E.g., to find the highest common factor of 4x3 12x2 + 11 x and 6x313x2 + 9x - 2.

6 x3-13x2+ 9x-2

2

4x3-12 x2+11x-3 12 x3-26 x2+18x-43

12 x3-36 x2+33x-9

5 10 x2-15x+5

Here the introduction of the factor 2 and the suppression of 5 evi

dently do not affect the degree of the highest common factor.

142. In practice, detached coefficients should be used whenever the problem warrants.

E.g., to find the highest common factor of

3x5y + 3x1y + 2 x3y — x2y — xy and 2x4 +9x3 + 9 x2 + 7x. Here x is evidently a factor of the highest common factor. It may therefore be suppressed and introduced later, thus shortening the work.

But y is a factor of the first only, and hence may be rejected entirely.

The problem then reduces to finding the highest common factor of 3x4 + 3x3 + 2 x2 - x 1 and 2 x3 + 9 x2 + 9x+7.

143. The work can often be abridged by noticing the difference between the two polynomials.

By the Remainder Theorem x - 1 is a factor of each expression, and x + 1 is not; .. x - 1 is the highest common factor of the expressions.

144. The highest common factor of three expressions cannot be of higher degree than that of any two; hence, the highest common factor of this highest common factor and of the third expression is the highest common factor of all three. Similarly, for any number of expressions.

EXERCISES. LVII.

Find the highest common factor of each of the following sets of expressions:

1. x3 − 2 x + 4, x2 + x3 + 4 x.

2. 2x+2x-4, x3-3x + 2.

3. x+4, x1 — 2 x3 + x2 + 2x-2.

A. x3-40x + 63, x* - 7 x3 +63 x 81. 5. x + y3, x1 — y*, x5 + x3y2 + x2y3 + y5.

6. x3 (6x+1)-x, 4x3- 2x (3x+2)+3. 7. . x 15 x2 + 28x12, 2x8 - 15 x + 14.

+10. 63 a 17 a3 + 17 a-3, 98 a2 + 34 a2 + 18.

x8

11. x2+4x-21, x2+ 20x +91, 2x+4x2 - 70 x. 12. 8 x 10 x3 +7 x2 - 2x, 6x5 - 11 x2+8x3- 2x2. 13. 9a24b2+ 4 bcc2, 2b2+ c2+3 ab -3 bc-3 ac. 14. (a - b) (a2 - c2) — (a — c) (a2 — b2), a5 — b3, ab — b2

ac + bc.

15. x3-10 (x2+3)+ 31 x, x2(x-11)+2 (19 x 20), x3-9x2+26 x 24.

16. a1b2+4a3b3 + 3 a2b1 — 4 ab2 — 4 b3, a5b + 3 a1b2 — a3b3 3 a2b4 4 a2b+4b8.

b2

17. 3 a2-7 ab +262 + 5 ac- 5 bc + 2 c2, 12 a2 - 19 ab

+5b2+11 ac - 11 be + 2 c2.

II. LOWEST COMMON MULTIPLE.

145. The integral algebraic multiple of lowest degree common to two or more algebraic expressions is called their lowest common multiple.

E.g., a2b3cd is the lowest common multiple of a2bc and ab3d.

Similarly, (a + b)2 (a − b) is the lowest common multiple of a2 — b2, b — a, and (a + b)2. For

1.

2.

3.

4. . either (a + b)2 (a − b) or (a + b)2 (b − a) contains the given expressions and is the common multiple of lowest degree.

The lowest common multiple of algebra must not be considered the same as the least common multiple when numerical values are assigned. E.g., the lowest common multiple of a + b and ab is (a+b) (a - b); but if a = 6 and b 4, the least common multiple of 6 + 4 and 6 − 4 is simply 6 + 4.

=

146. So far as the algebraic multiple is concerned, numerical factors are not usually considered.

E.g., a2b3c is the lowest common multiple of 2 ab3c, a2b, and 15 ab.

The lowest common multiple is used in reducing fractions to fractions having a lowest common denominator.

147. Factoring method. The lowest common multiple is usually found by the inspection of factors.

E.g., to find the lowest common multiple of x2 12x+27, x2 + x - 12, and 15 – 2 x − x2.

1.

2.

3.

4. ..± (x − 3) (x + 4) (x + 5) (x — 9) is the lowest common multiple.

In practice, the result should be left in the factored form.