Problem 17 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &x-1\\\ &x-2\\\ &(x-1)(x-2) \end{aligned}$$

Step-by-Step Solution

Answer

The Least Common Multiple (LCM) of the polynomials \(x-1\), \(x-2\), and \((x-1)(x-2)\) is \((x-1)(x-2)\).

Step 1: Identify the polynomials

The given polynomials are \(x-1\), \(x-2\), and \((x-1)(x-2)\).

Step 2: Recognize the third polynomial

Notice that the third polynomial is the product of the first two polynomials. This means that it is the polynomial of the lowest degree divisible by both \(x-1\) and \(x-2\). Therefore, the least common multiple (LCM) of the given polynomials is the third polynomial itself.

Step 3: Confirm the LCM

To ensure that \((x-1)(x-2)\) is the LCM, one could perform the division of the third polynomial by the first two polynomials. In this case, it is clear that the divisions would result in no remainder, confirming that \((x-1)(x-2)\) is the LCM of the given polynomials.

Key Concepts

Polynomial Factorization

Polynomial factorization is an essential concept, especially when working with polynomials to find solutions or simplifications. It involves expressing a polynomial as a product of simpler polynomials. These simpler polynomials are referred to as factors, and this technique is akin to breaking down a number into its prime factors.
By factoring polynomials, students can simplify math problems and understand more complex polynomial relationships. For example, the polynomial ul- \(x^2 - 3x + 2\) can be factored into \((x-1)(x-2)\).\[/ul\]Here, \(x-1\) and \(x-2\) are the factors of the given polynomial. Understanding polynomial factorization is vital because it lays the groundwork for other operations, such as finding the least common multiple (LCM) or solving polynomial equations. Factoring also helps identify the roots of the polynomial, which are the values of \(x\) that make the polynomial equal to zero.

Lowest Common Multiple (LCM)

The lowest common multiple (LCM) of polynomials is the smallest polynomial that is divisible by the given polynomials. When you find the LCM, you are essentially looking for the simplest form of a polynomial that can be divided by the original set of polynomials without leaving a remainder.
In the given exercise, we identify that the polynomial \((x-1)(x-2)\) is the LCM of \(x-1\), \(x-2\), and \((x-1)(x-2)\). Because:

- It is the product of the factors of the other two polynomials.- It is of the lowest degree that can be divided by each individual polynomial without remainder.

Finding the LCM is crucial in solving polynomial equations and is often necessary for adding or subtracting fractions with polynomial denominators. The process of finding the LCM can be simplified if the original polynomials are factored beforehand.

Division of Polynomials

Division of polynomials is the process used to determine whether one polynomial can be divided by another without leaving a remainder. It is very similar to the long division used with numbers.
In this context, it verifies that a polynomial is the lowest common multiple (LCM) through division. If one polynomial can divide another without leaving a remainder, it is a multiple of the other polynomial. This method was briefly demonstrated in the given solution.

- The polynomial \((x-1)(x-2)\) was tested against the polynomials \(x-1\) and \(x-2\).

Since dividing \((x-1)(x-2)\) by \(x-1\) results in \(x-2\), and dividing it by \(x-2\) results in \(x-1\), the division confirms that \((x-1)(x-2)\) is indeed the LCM. Understanding polynomial division is not only crucial for confirming LCMs, but it is also a foundation for further polynomial manipulations in algebra.