Chapter 19: Highest Common Factor and Lowest Common Multiple

In this chapter, we apply our factoring skills to find the greatest shared factor between expressions and the smallest expression that is a multiple of several others. These tools are the "gears" that allow us to add and subtract algebraic fractions with different denominators.

19.1 Highest Common Factor (H.C.F.)

The Highest Common Factor of two or more algebraic expressions is the expression of highest degree and largest numerical coefficient that will divide each of them without a remainder.

To find the H.C.F.: Factor each expression completely. The H.C.F. is the product of all the common factors, each taken with its lowest exponent.

Example:
Find the H.C.F. of: 4x²y³ and 6x³y²

Step 1: Factor the coefficients:
4 = 2², 6 = 2 × 3. Common numerical factor is 2.

Step 2: Compare variable powers:
For x: x² and x³ → lowest power is x².
For y: y³ and y² → lowest power is y².

Result: 2x²y²

19.2 Lowest Common Multiple (L.C.M.)

The Lowest Common Multiple of two or more expressions is the expression of lowest degree and smallest numerical coefficient that is exactly divisible by each of them.

To find the L.C.M.: Factor each expression completely. The L.C.M. is the product of all the different factors, each taken with its highest exponent.

Example:
Find the L.C.M. of: x² - 1 and (x - 1)²

Step 1: Factor completely:
x² - 1 = (x + 1)(x - 1)
(x - 1)² = (x - 1)(x - 1)

Step 2: Collect all unique factors at their highest powers:
The factors are (x + 1) and (x - 1).
Highest power of (x + 1) is 1.
Highest power of (x - 1) is 2.

Result: (x + 1)(x - 1)²

19.3 H.C.F. by Successive Division

For very large expressions that are difficult to factor by sight, Wentworth teaches the method of Successive Division (similar to the Euclidean Algorithm). This involves dividing the higher-degree expression by the lower-degree expression and using the remainder as the new divisor until no remainder remains.

Doctor's Pro-Tip: Finding the L.C.M. is the most important skill here because it is exactly what you need to find a Common Denominator when adding algebraic fractions in the next chapter.

Chapter 19 exercises