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nydus/The First Steps in AlgebraPublic

This textbook introduces the fundamental concepts of algebra, including definitions of units, quantities, and number symbols. It explains the use of letters to represent numbers and outlines the signs for basic mathematical operations.

Page 91 of 227

Table of Contents

VIII.

Find the product of all the common factors, taking each factor the least number of times it occurs in any of the given expressions.

The highest common factor in Algebra corresponds to the greatest common measure, or greatest common divisor in Arithmetic. We cannot apply the terms greatest and least to an algebraic expression in which particular values have not been given to the letters contained in the expression. Thus $a$ is greater than $a^{2}$ , if $a$ stands for $\frac{1}{4}$ .

Exercise 40.

Find the H. C. F. of:

$330$ and $546$ .

$20 x^{3}$ and $15 x^{4}$ .

$42 a x^{2}$ and $60 a^{2} x$ .

$35 a^{2} b^{2}$ and $49 a b^{3}$ .

$28 x^{4}$ and $63 y^{4}$ .

$54 a^{2} b^{2}$ and $56 a^{3} b^{3}$ .

$x^{3} + 3 x^{2} y$ and $x^{3} + 27 y^{3}$ .

$x^{2} + 3 x$ and $x^{2} - 9$ .

$2 a x^{3} + x^{3}$ and $8 a^{3} + 1$ .

$(x + y)^{2}$ and $x^{2} - y^{2}$ .

$a^{3} + a^{2} x$ and $a^{2} - x^{2}$ .

$a^{2} - 4 b^{2}$ and $a^{2} + 2 a b$ .

$x^{2} - 1$ and $x^{2} + 2 x - 3$ .

$x^{2} + 5 x + 6$ and $x^{2} + 4 x + 3$ .

$x^{2} - 9 x + 18$ and $x^{2} - 10 x + 24$ .

$x^{3} + 1$ and $x^{2} - x + 1$ .

$x^{2} - 3 x + 2$ and $x^{2} - 4 x + 3$ .

$x^{2} - 3 x y + 2 y^{2}$ and $x^{2} - 2 x y + y^{2}$ .

$x^{2} - 4 x - 5$ and $x^{2} - 25$ .

$(a - b)^{2} - c^{2}$ and $a b - b^{2} - b c$ .

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