Page:Elementary algebra (1896).djvu/110

92 Rh CHAPTER XL

Highest Common Factor.

110. Definition. The Highest Common Factor of two or more algebraic expressions is the expression of highest dimensions (Art. 29) which divides each of them without remainder.

The abbreviation H. C. F. is sometimes used instead of the words highest common factor.

SIMPLE EXPRESSIONS.

111. The H. C. F. can be written by inspection.

Ex. 1. The highest common factor of a4, a3, a2, a^6 is a^2.

Ex. 2. The highest common factor of a^3b^4, ab^5c^2, a^2b^7c is ab^4 ; for a is the highest power of a that will divide a^3, a, a^2 ; b^4 is the highest power of b that will divide b^4, b^5, b^7; and c is not a common factor.

112. If the expressions have numerical coefficients, find by Arithmetic their greatest common measure, and prefix it as a coefficient to the algebraic highest common factor.

Ex. The highest common factor of 21 a4x3y, 35a2x4y, 28a3xy4 is 7 a^2xy ; for it consists of the product of

(1) the numerical greatest common measure of the coefficients ;

(2) the highest power of each letter which divides every one of the given expressions.

EXAMPLES XI. a.

Find the highest common factor of

1. 4ab2, 2a2b. 3. 6xy2z, 8x^2y^3z^2 5. 5a3b3, 15 abc3. 2. 3x^2y^2, x8y2. 4. abc, ab^2c. 6. 9x2y2z2, 12 xy^3z.