Question

Find the L.CM of the given polynomials. $$ x^{2}-4, x^{2}-x-2 $$

Short answer

The L.C.M. of the given polynomials is \( (x - 2)(x + 2)(x + 1) \).

Step-by-step solution

- Factorize the first polynomial

The first polynomial is given as \( x^2 - 4 \). This is a difference of squares, which can be factored as: \( x^2 - 4 = (x - 2)(x + 2) \).

- Factorize the second polynomial

The second polynomial is given as \( x^2 - x - 2 \). To factor this, we look for two numbers that multiply to -2 and add to -1. These numbers are -2 and 1. Therefore, \( x^2 - x - 2 = (x - 2)(x + 1) \).

- Find the Least Common Multiple (L.C.M.)

The factorizations of the polynomials are: \( x^2 - 4 = (x - 2)(x + 2) \)\( x^2 - x - 2 = (x - 2)(x + 1) \). To find the L.C.M., take each distinct factor at its highest power that appears in any of the factorizations.So, the L.C.M. is:\( (x - 2)(x + 2)(x + 1) \).

Key concepts

Factorization of Polynomials

Polynomials can often be simplified through a process called factorization. Factorizing a polynomial involves breaking it down into simpler polynomials called factors that, when multiplied together, produce the original polynomial. To better understand, consider the polynomial \(x^2 - x - 2\). We need to find two numbers that multiply to give -2 and add up to -1. These numbers are -2 and 1. Hence, \(x^2 - x - 2\) becomes \((x - 2)(x + 1)\).

Another key instance in factorization is the difference of squares, where you can factor a polynomial like \(x^2 - 4\) into \((x - 2)(x + 2)\).
Always be on the lookout for common patterns, such as quadratic equations with leading coefficients of 1 and constant terms.

Difference of Squares

The difference of squares is a specific type of polynomial factorization that applies when you have a polynomial in the form of \(a^2 - b^2\). To factor this, you use the identity:

\[a^2 - b^2 = (a - b)(a + b)\]

In the given exercise, \(x^2 - 4\) is a difference of squares because it can be written as \((x)^2 - (2)^2\). Applying the difference of squares identity results in \(x^2 - 4 = (x - 2)(x + 2)\).

Recognizing this pattern is crucial for simplifying polynomials and solving equations.

Least Common Multiple

The least common multiple (LCM) of two or more polynomials is the smallest polynomial that is a multiple of each of the original polynomials. To find the LCM, factor each polynomial and then take each distinct factor at its highest power that appears in any of the factorizations.

Consider the polynomials from the exercise:
- \(x^2 - 4\) factors as \((x - 2)(x + 2)\)
- \(x^2 - x - 2\) factors as \((x - 2)(x + 1)\)

The distinct factors are \(x - 2\), \(x + 2\), and \(x + 1\). Therefore, the LCM is obtained by multiplying these factors together:

\[(x - 2)(x + 2)(x + 1)\]

Understanding how to find the LCM of polynomials allows for efficient simplification and combination of polynomial expressions.