Problem 56 - Algebra 2

Question

Find the least common multiple of \(x^{2}-1\) and \(x^{2}-x\). A. \(x-1\) B. \(x(x-1)(x+1)\) C. \(x(x-1)^{2}(x+1)\) D. \((x-1)^{2}(x+1)^{2} x^{2}\)

Step-by-Step Solution

Answer

The least common multiple of \(x^{2}-1\) and \(x^{2}-x\) is \(x(x-1)(x+1)\), so the correct answer is option B.

Step 1: Factorise the expressions

The first step is to factorise the two given expressions. \(x^{2}-1\) can be factorised as \((x-1)(x+1)\) using difference of squares, and \(x^{2}-x\) can be factorised as \(x(x-1)\), by taking x as a common factor.

Step 2: Find the LCM

The LCM of two algebraic expressions is the product of the highest powers of all factors that appear in either of the expressions. Here, the only factors that appear are \(x, (x-1)\), and \((x+1)\). Thus, the LCM is the product of these three factors, which is \(x(x-1)(x+1)\).

Step 3: Match the LCM with answer options

Lastly, match the LCM found in step 2 with the given answer options. The LCM matches with answer option B. So, the correct answer is option B.

Key Concepts

Factoring Algebraic Expressions

Factoring algebraic expressions is a crucial skill in algebra that involves rewriting an expression as a product of simpler expressions, called factors. This process can significantly simplify solving equations and finding the least common multiple (LCM) of algebraic terms.

In our exercise, we start by factoring two expressions: \(x^2 - 1\) and \(x^2 - x\). When factoring, always look for common factors or recognizable patterns like the difference of squares or trinomial squares.

For instance:

• For \(x^2 - 1\), observe it can be rewritten as a difference of squares: \((x - 1)(x + 1)\).
• For \(x^2 - x\), factor by taking out the greatest common factor (GCF), which is \(x\), resulting in \(x(x - 1)\).

Understanding these basic techniques of factoring helps in simplifying expressions, solving equations, and working more efficiently with algebraic expressions.

Difference of Squares

The difference of squares is a specific type of algebraic factoring and forms part of many factorization problems. This method is used to simplify expressions of the form \(a^2 - b^2\), which can be factored into \((a - b)(a + b)\).

In analyzing the expression \(x^2 - 1\) from our exercise, notice this is a perfect example of a difference of squares because:

• It is expressed as \(x^2 - 1^2\).
• Therefore, it factors into \((x - 1)(x + 1)\).

The beauty of this method is its simplicity and effectiveness. Once identified, the difference of squares dramatically simplifies complex expressions or problems, making it easier to work with them within equations or when finding the least common multiple.

Highest Power of Factors

When determining the least common multiple of algebraic expressions, identifying the highest power of all factors present in the expressions is key.

The LCM must include each factor at its greatest occurrence in any of the expressions involved. This ensures that the LCM encompasses every part of the original expressions.

Consider the expressions \(x^2 - 1\) and \(x^2 - x\) in our task:

• Factors are \(x, (x-1), (x+1)\).
• The LCM needs \(x\), \((x-1)\), and \((x+1)\).

Thus, the LCM here becomes \(x(x-1)(x+1)\), which is the combination of these factors at their highest powers. Mastery of the highest power of factors helps tremendously in simplifying and solving multiple algebraic problems efficiently.