Question

Find the L.CM of the given polynomials. $$ x^{3}-x, x^{2}-x $$

Short answer

The LCM of \(x^3 - x\) and \(x^2 - x\) is \(x(x-1)(x+1)\).

Step-by-step solution

Factor the Polynomials

First, factor each polynomial. For the polynomial \(x^3 - x\), factor out the common term \(x\): \(x(x^2 - 1)\). Then factor \(x^2 - 1\) as a difference of squares: \(x(x-1)(x+1)\). For the polynomial \(x^2 - x\), factor out the common term \(x\): \(x(x-1)\).

Identify the Common and Unique Factors

List all unique factors, including repeated ones where necessary: \(x(x-1)(x+1)\) and \(x(x-1)\). The factors are \(x, (x-1), (x+1)\).

Determine the Highest Power of Each Factor

For each unique factor, identify the highest power it appears in any polynomial. Here, \(x, (x-1), (x+1)\) each appear to the power of one.

Multiply the Factors

Multiply the highest power of each of the unique factors to find the LCM: \(x \times (x-1) \times (x+1)\).

Simplify the Expression

Simplify the multiplication to get the LCM: \(x(x-1)(x+1)\).

Key concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components called factors. These factors, when multiplied together, give back the original polynomial. For example, the polynomial \( x^3 - x \) can be factored as follows:
First, identify common factors in the polynomial. In \( x^3 - x \), the common factor is \( x \):
\[ x(x^2 - 1) \]. Next, recognize if the remaining component \( x^2 - 1 \) can be factored further. Indeed, it can, using the difference of squares technique:
\[ x(x-1)(x+1) \].
Remember, the initial step in factoring is identifying common factors and then employing techniques like the difference of squares, factoring by grouping, or recognizing patterns like sum/difference of cubes to break down the polynomials further.

Difference of Squares

The difference of squares is a special factoring technique for polynomials where two squared terms are subtracted. It follows this formula: \( a^2 - b^2 = (a-b)(a+b) \).
Recognizing a difference of squares is key to factoring certain polynomials easily. For example, \( x^2 - 1 \) is a difference of squares. Here, \( a = x \) and \( b = 1 \), so:
\[ x^2 - 1 = (x-1)(x+1) \].
Using this technique simplifies the polynomial, making it easier to work with in subsequent steps.

Least Common Multiple

The Least Common Multiple (LCM) of polynomials is the smallest polynomial in terms of degree that can be divided evenly by the given polynomials. To find the LCM:
- First, factor each polynomial completely.
- Next, list all unique factors, including their highest powers in any polynomial.
For our example, factoring gives us \( x^3 - x = x(x-1)(x+1) \) and \( x^2 - x = x(x-1) \). Here, the unique factors are \( x, (x-1), (x+1) \).
Finally, multiply the highest power of each unique factor to form the LCM:
\[ x \times (x-1) \times (x+1) = x(x-1)(x+1) \].
This product represents the LCM, ensuring that it encompasses all the factors from the original polynomials.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial and then combining like terms. This is essential in finding the LCM.
Given our factors \( x, (x-1), (x+1) \), multiplying them looks like this:
Multiply \( x \) with \( (x-1) \):
\[ x \times (x-1) = x^2 - x \].
Then, multiply the resulting expression by \( (x+1) \):
\[ (x^2 - x) \times (x+1) \].
To multiply, use the distributive property:
\[ x^2 \times x + x^2 \times 1 - x \times x - x \times 1 \].
This simplifies to:
\[ x^3 + x^2 - x^2 - x = x^3 - x \].
Thus, polynomial multiplication streamlines the process of finding the LCM by breaking it down into smaller, more manageable steps.