MCD e mcm di polinomi

With polynomials the exact same story as with numbers repeats: the role of the prime factors is played by the irreducible factors of the polynomial (the “bricks” that no longer break apart, such as $(x-1)$, $(x+1)$, $x$, $x^2+1$). Once the polynomials are factored, the GCD and LCM are read off with the same two methods — sets of factors and table.

Property — GCD and LCM of polynomials

Once the polynomials are factored into irreducible factors:

• GCD: product of the common factors, each with the smaller exponent;
• LCM: product of all the factors, each with the larger exponent.

The LCM of the denominators is what you need in order to add or subtract algebraic fractions: it is the smallest common denominator.

Example

Let us factor three polynomials: $A = x^2-1 = (x-1)(x+1), \qquad B = x^2+2x+1 = (x+1)^2, \qquad C = x^3-x = x(x-1)(x+1).$

Table method — one column for each irreducible factor, the exponent in the cell:

	$x$	$(x-1)$	$(x+1)$
$A$	$0$	$1$	$1$
$B$	$0$	$0$	$2$
$C$	$1$	$1$	$1$
min	$0$	$0$	$1$
max	$1$	$1$	$2$

$\text{MCD}(A,B,C) = (x+1)^{\min} = (x+1), \qquad \text{mcm}(A,B,C) = x\,(x-1)\,(x+1)^2.$

The only factor present in all three rows is $(x+1)$: that is the GCD. The LCM gathers every factor with the largest exponent.

Method of sets of factors

The same result with a Venn diagram, thinking of each polynomial as the set of its factors — the intersection gives the GCD, the union the LCM. With $A=(x-1)(x+1)$ and $B=(x+1)^2$:

$\text{MCD}(A,B) = (x+1), \qquad \text{mcm}(A,B) = (x-1)(x+1)^2.$

Why it is needed: the common denominator

To compute $\frac{1}{x^2-1} + \frac{1}{x^2+2x+1} = \frac{1}{(x-1)(x+1)} + \frac{1}{(x+1)^2}$ one takes as common denominator the LCM $=(x-1)(x+1)^2$: $= \frac{(x+1) + (x-1)}{(x-1)(x+1)^2} = \frac{2x}{(x-1)(x+1)^2}.$

Exactly as with numbers, where for $\tfrac{2}{15}-\tfrac{3}{10}$ one uses $\text{mcm}(15,10)=30$.

Links

Topics: Frazioni algebriche
Concepts: Massimo comun divisore (MCD) · Minimo comune multiplo (mcm) · Denominatore comune
Skills: Scomporre