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Chapter 9: Problem 4

Find the least common multiple of each pair of polynomials. 9\((x+2)(2 x-1)\) and 3\((x+2)\)

Short Answer

Expert verified
The least common multiple of the polynomials \(9(x+2)(2 x-1)\) and \(3(x+2)\) is \(9(x+2)(2 x-1)\).

Step by step solution

01

Identify the polynomials

The given polynomials are \(9(x+2)(2 x-1)\) and \(3(x+2)\)
02

Identify the common and uncommon factors

The common factor in the two polynomials is \((x+2)\). But, the first polynomial \(9(x+2)(2 x-1)\) also contains an additional factor \((2 x-1)\). Meanwhile, the factor 9 in the first polynomial is composed of \(3 \times 3\) which includes 3, a constant factor in the second polynomial.
03

Formulate the LCM

To obtain the LCM of the two polynomials, pick the common factor, \((x+2)\). Additionally, include the highest power of all other factors from both the polynomials. Here, the first polynomial has an extra factor \((2 x-1)\). Thus, include this factor in the LCM. Similarly, among the constants 3 and 9, pick the one with the highest power, which is 9. So, the LCM becomes \(9(x+2)(2 x-1)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Factorization
When working with polynomials, a key tool is polynomial factorization. This process involves breaking down a polynomial into a product of its simplest parts, called factors. Factoring is like finding the building blocks of polynomials.
  • It helps in simplifying polynomial expressions.
  • Makes it easier to find solutions to polynomial equations.
  • Essential for calculating the least common multiple of polynomial expressions.

For instance, the polynomial \(9(x+2)(2x-1)\) is already factored into its simplest linear form, where \(9\) and the binomials \((x+2)\) and \((2x-1)\) are the factors. Identifying these factors is the first step toward solving more complex problems, such as finding the least common multiple of polynomial pairs.
LCM of Expressions
The Least Common Multiple (LCM) of expressions is an important concept when dealing with polynomials. The LCM of two polynomial expressions is the smallest polynomial that is divisible by both original expressions. It is particularly useful when trying to combine fractions with polynomial denominators or equalize polynomials to a common form.
  • The LCM incorporates all the unique factors from the expressions.
  • It takes the highest powers of these factors present in any of the polynomials.

In our example with polynomials \(9(x+2)(2x-1)\) and \(3(x+2)\), the LCM must include the factor \((x+2)\), which is common, as well as \((2x-1)\) and the constant factor 9, since 9 is the highest power among the numeric factors. Hence, the LCM is \(9(x+2)(2x-1)\). This ensures that the LCM is divisible by both polynomials accurately.
Common Factors in Polynomials
Common factors in polynomials are those factors that appear in each polynomial when factored. They are crucial in the process of finding the Least Common Multiple and simplifying expressions.
  • Determining common factors helps to simplify complex polynomial expressions.
  • The common factor is simply the factor repeated in all expressions.
  • In our example, the common factor is \((x+2)\).

Recognizing common factors allows us to perform operations like addition, subtraction, and solving equations more efficiently. By identifying \((x+2)\) as a common factor, we lay the groundwork for further operations such as determining the factor with the greatest power when calculating the LCM. This ability to spot commonalities in polynomial expressions is a valuable skill in algebra.

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Most popular questions from this chapter

Open-Ended. Write three rational expressions that simplify to \(\frac{x}{x+1}\)

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x-1}{(3 x+2)(x+1)} $$

Multiply or divide. State any restrictions on the variable. $$ \frac{2 x^{2}-6 x}{x^{2}+18 x+81} \cdot \frac{9 x+81}{x^{2}-9} $$

Evaluate each logarithm. $$ \log _{16} 8 $$

Multiply or divide. State any restrictions on the variable. $$ \frac{2 x^{2}+5 x+2}{4 x^{2}-1} \cdot \frac{2 x^{2}+x-1}{x^{2}+x-2} $$
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