VIII.

$x^{2} + x y - 2 y^{2}$ and $x^{2} + 5 x y + 6 y^{2}$ .

$x^{2} + 7 x y + 12 y^{2}$ and $x^{2} + 3 x y - 4 y^{2}$ .

$x^{3} - 8 y^{3}$ and $x^{2} + 2 x y + 4 y^{2}$ .

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

$1 - 5 a + 6 a^{2}$ and $1 - 7 a + 12 a^{2}$ .

$x^{2} - 8 x y + 7 y^{2}$ and $x^{2} - 3 x y - 28 y^{2}$ .

$8 a^{3} + b^{3}$ and $4 a^{2} + 4 a b + b^{2}$ .

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

123. Common Multiples.

A common multiple of two or more integral numbers is a number which is exactly divisible by each, of the numbers.

A common multiple of two or more expressions is an expression which is exactly divisible by each of the expressions.

The lowest common multiple of two or more numbers is the least number that is exactly divisible by each of the given numbers.

The lowest common multiple of two or more expressions is the expression of lowest degree that is exactly divisible by each of the given expressions.

We use L. C. M. for "lowest common multiple."

To find the lowest common multiple of two or more algebraic expressions.

Find the L. C. M. of $42 a^{3} b^{2}$ , $30 a^{2} b^{4}$ , and $66 a b^{3}$ .

$42 a^{3} b^{2} & = 2 \times 3 \times 7 \times a^{3} \times b^{2}; 30 a^{2} b^{4} & = 2 \times 3 \times 5 \times a^{2} \times b^{4}; 66 a b^{3} & = 2 \times 3 \times 11 \times a \times b^{3} .$

The L. C. M. must evidently contain each factor the greatest number of times that it occurs in any expression.

$∴ L.\,C.\,M. & = 2 \times 3 \times 7 \times 5 \times 11 a^{3} \times b^{4}, & = 2310 a^{3} b^{4} .$

Find the L. C. M. of $4 x^{2} - 4 x - 80$ and $2 x^{2} - 18 x + 40$ .

$3 & 4 x^{2} - & 4 x & - 80 & & = 4 (x^{2} - x - 20) = 4 (x - 5) (x + 4); & 2 x^{2} - & 18 x & + 40 & & = 2 (x^{2} - 9 x + 20) = 2 (x - 5) (x - 4) . ∴ & \text{L.\,C.\,M.} & & & & = 4 (x - 5) (x + 4) (x - 4) .$

find the L. C. M. of two or more expressions, therefore,

Resolve each expression into its simplest factors.