Question

Find the LCM of each set of polynomials. $$ x^{2}-2 x, x^{2}-4 $$

   Find the LCM of each set of polynomials.
$$
x^{2}-2 x, x^{2}-4
$$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 8, Problem 3

Instant Answer

Solved by Expert Ashley High

03/18/2020

Step 1

The first polynomial $x^{2}-2 x$ can be factored by taking out a common factor of $x$, which gives us $x(x-2)$. The second polynomial $x^{2}-4$ is a difference of squares and can be factored into $(x-2)(x+2)$. Show more…

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Find the LCM of each set of polynomials. $$ x^{2}-2 x, x^{2}-4 $$

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Key Concepts

Polynomial Factorization

This concept involves expressing a polynomial as a product of its factors. It is a crucial technique for simplifying polynomial expressions and solving equations, as well as for identifying structures such as common factors or factorable patterns like the difference of squares.

Least Common Multiple (LCM) in Polynomial Rings

The LCM of polynomials is the smallest polynomial (up to multiplication by a nonzero constant) that is divisible by each polynomial in a given set. It is analogous to the LCM for integers and is used to combine polynomial expressions, especially when adding rational expressions with different denominators.

Identifying Common Factors

Identifying common factors in polynomial expressions is an important step in factoring and simplification. By extracting the greatest common divisor from each polynomial, one can reduce the problem to factoring simpler expressions, which aids in finding the LCM effectively.

Difference of Squares Factoring

The difference of squares is a special factoring pattern where a polynomial of the form a^2 - b^2 can be factored into (a - b)(a + b). Recognizing this pattern is useful for quickly breaking down polynomials and simplifying them, which is often necessary when determining common factors and the LCM.

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