Problem 66 - College Algebra with Modeling and Visualization

Question

Find the least common multiple. $$ x^{2}-4, x^{2}-4 x+4 $$

Step-by-Step Solution

Answer

The LCM is \((x - 2)^2(x + 2)\).

Step 1: Factorize the First Expression

The first expression is \(x^2 - 4\). This is a difference of squares which can be factorized as: \[x^2 - 4 = (x - 2)(x + 2)\]

Step 2: Factorize the Second Expression

The second expression is \(x^2 - 4x + 4\). This is a perfect square trinomial, which can be factorized as: \[x^2 - 4x + 4 = (x - 2)^2\]

Step 3: Determine All Factors

From Step 1, we have the factors \((x - 2)\) and \((x + 2)\). From Step 2, we have the factor \((x - 2)\) repeated, forming \((x - 2)^2\).

Step 4: Write Down the Least Common Multiple

The least common multiple (LCM) is found by taking the highest power of each distinct factor from the two expressions. The distinct factors are \((x - 2)\) and \((x + 2)\). The LCM is: \[(x - 2)^2(x + 2)\]

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebra. It refers to expressing an equation in the form of \(a^2 - b^2\). This type of expression can be factorized into two binomials: \((a - b)(a + b)\). This factorization is useful because it simplifies complex algebraic expressions and aids in finding the least common multiple (LCM).

In the original exercise, the expression \(x^2 - 4\) is a perfect example of a difference of squares. Here, it can be rewritten as \((x)^2 - (2)^2\), which means \(a = x\) and \(b = 2\).

Hence, it can be expressed as the product of two simpler expressions:

• \(x - 2\)
• \(x + 2\)

Understanding how to identify and factor out the difference of squares allows for simplification of expressions, eventually helping in finding the LCM.

Perfect Square Trinomial

A perfect square trinomial is another special form of a polynomial. It usually looks like \(a^2 ± 2ab + b^2\), and it results from squaring a binomial.

To factor a perfect square trinomial, you look for two key features:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

In the problem we have, the expression \(x^2 - 4x + 4\) is a perfect square trinomial because it can be associated with \((x - 2)^2\).

Let's break it down:
The expression \(x^2\) is a square of \(x\), \(4\) is a square of \(2\), and the middle term \(-4x\) equates to \(2(-x)(2)\). This symmetry confirms that the expression is of the form \(a^2 - 2ab + b^2 = (a - b)^2\).

Recognizing and factorizing perfect square trinomials ease the process of finding the least common multiple, as it simplifies the polynomial to its core components.

Factorization

Factorization is the process of breaking down a complex mathematical expression into simpler, more manageable components called factors. This process is crucial in solving algebraic equations, particularly when finding the least common multiple (LCM).

In the original exercise, factorization is applied to both the difference of squares and the perfect square trinomial.
The expression \(x^2 - 4\) is factorized into \((x - 2)(x + 2)\), while \(x^2 - 4x + 4\) is factorized into \((x - 2)^2\).

By determining these factors, the LCM is then derived from the highest powers of each distinct factor. In this exercise, the factors are:

• \((x - 2)\) with a highest power of 2
• \((x + 2)\)

Thus, the LCM of the two polynomial expressions is \((x - 2)^2(x + 2)\).

Embracing factorization not only simplifies expressions but also lays the groundwork for solving more complex algebraic equations efficiently.