Least Common Multiple of Polynomials

The least common multiple (LCM) of two or more polynomials is the polynomial of lowest possible degree that is divisible by all of them.

How to Determine the LCM of Polynomials

To compute the least common multiple of several polynomials:

1. Express each polynomial in its irreducible factored form.
2. Multiply together all distinct factors - both shared and unique - each taken only once, with the highest exponent that appears among the polynomials.

Note. This process closely mirrors how we find the LCM of two or more integers: decompose into irreducible factors, then take the product of all factors, each raised to its greatest exponent.

A Practical Example

Consider the following polynomials:

$$ A(x): 6x^3 $$

$$ B(x): 3x^4 - 27x^2 $$

$$ C(x): 2x^3 + 6x^2 $$

We begin by factoring each polynomial completely:

$$ A(x): 2 \cdot 3 \cdot x^3 $$

$$ B(x): 3 \cdot x^2 \cdot (x - 3) \cdot (x + 3) $$

$$ C(x): 2 \cdot x^2 \cdot (x + 3) $$

Explanation. The factorization of A(x) is straightforward: $$ A(x): 6x^3 $$ Decomposing the constant 6 gives: $$ A(x): 2 \cdot 3 \cdot x^3 $$. Next, for B(x): $$ B(x): 3x^4 - 27x^2 $$ Factor out the greatest common monomial factor, 3: $$ B(x): 3 \cdot (x^4 - 9x^2) $$ Then factor out \( x^2 \): $$ B(x): 3 \cdot x^2 \cdot (x^2 - 9) $$ Recognize a difference of squares: $$ B(x): 3 \cdot x^2 \cdot (x - 3)(x + 3) $$ Finally, for C(x): $$ C(x): 2x^3 + 6x^2 $$ Factor out 2: $$ C(x): 2 \cdot (x^3 + 3x^2) $$ Then factor out \( x^2 \): $$ C(x): 2 \cdot x^2 \cdot (x + 3) $$

$$ \begin{vmatrix} A(x) & 2 & 3 & x^3 \\ B(x) & & 3 & x^2 & x-3 & x+3 \\ C(x) & 2 & & x^2 & & x+3 \end{vmatrix} $$

Now identify all distinct and shared irreducible factors across the three polynomials.

From these, take each factor only once, selecting its highest degree among the occurrences.

$$ \begin{vmatrix} A(x) & \color{red}2 & \color{red}3 & \color{red}{x^3} \\ B(x) & & 3 & x^2 & \color{red}{x-3} & \color{red}{x+3} \\ C(x) & 2 & & x^2 & & x+3 \end{vmatrix} $$

Multiplying all selected factors gives the LCM:

$$ \text{LCM}[A(x), B(x), C(x)] = 2 \cdot 3 \cdot x^3 \cdot (x - 3) \cdot (x + 3) $$

This polynomial is divisible by each of A(x), B(x), and C(x).

Example 2

Now let’s analyze another set of polynomials:

$$ A(x): 3x^2y + 3xy^2 $$

$$ B(x): 6x^3 + 6x^2y $$

$$ C(x): 2x^2y^2 + 2xy^3 $$

We factor each expression completely:

$$ A(x): 3 \cdot x \cdot y \cdot (x + y) $$

$$ B(x): 2 \cdot 3 \cdot x^2 \cdot (x + y) $$

$$ C(x): 2 \cdot x \cdot y^2 \cdot (x + y) $$

Explanation. Let’s start with A(x): $$ A(x): 3x^2y + 3xy^2 $$ Factor out 3: $$ A(x): 3 \cdot (x^2y + xy^2) $$ Then factor out \( x \): $$ A(x): 3 \cdot x \cdot (xy + y^2) $$ And finally factor out \( y \): $$ A(x): 3 \cdot x \cdot y \cdot (x + y) $$ Now B(x): $$ B(x): 6x^3 + 6x^2y $$ Factor out 6: $$ B(x): 6 \cdot (x^3 + x^2y) $$ Then rewrite as: $$ B(x): 2 \cdot 3 \cdot x^2 \cdot (x + y) $$ And C(x): $$ C(x): 2x^2y^2 + 2xy^3 $$ Factor out 2: $$ C(x): 2 \cdot (x^2y^2 + xy^3) $$ Then factor out \( x \): $$ C(x): 2 \cdot x \cdot (xy^2 + y^3) $$ Finally factor out \( y^2 \): $$ C(x): 2 \cdot x \cdot y^2 \cdot (x + y) $$

$$ \begin{vmatrix} A(x) & & 3 & x & y & x+y \\ B(x) & 2 & 3 & x^2 & x+y \\ C(x) & 2 & & x & y^2 & x+y \end{vmatrix} $$

We now collect all distinct and shared irreducible factors.

$$ \begin{vmatrix} A(x) & & \color{red}3 & x & y & \color{red}{x+y} \\ B(x) & \color{red}2 & 3 & \color{red}{x^2} & & x+y \\ C(x) & 2 & & x & \color{red}{y^2} & x+y \end{vmatrix} $$

Take each factor only once, using its highest degree across all polynomials, and multiply them together.

Thus, the least common multiple (LCM) is:

$$ \text{LCM}[A(x), B(x), C(x)] = 2 \cdot 3 \cdot x^2 \cdot y^2 \cdot (x + y) $$

This polynomial is divisible by A(x), B(x), and C(x).

And so on.