Problem 9 - Prealgebra and Introductory Algebra

Question

Find the LCM of the polynomials. $$\begin{aligned} &9 x(x+2)\\\ &12(x+2)^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The LCM of the two polynomials is \(36x^2(x+2)^2\).

Step 1: Simplify Polynomials

First, simplify both polynomials by multiplying out if it's needed. In this case, the first one \(9x(x+2)\) simplifies to \(9x^2+18x\). The second one \(12(x+2)^2\) simplifies to \(12x^2+48x+48\).

Step 2: Factor the Polynomials

Because the polynomials are now simplified, they can be factored. The first polynomial, \(9x^2+18x\), factors into \(9x(x+2)\). The second polynomial can't be further factored beyond its simplified form \(12x^2+48x+48\).

Step 3: Identify the Least Common Multiple (LCM)

The LCM is determined by taking each factor the greatest number of times it occurs in either polynomial. This gives \(LCM = 36x^2(x+2)^2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like finding what you need to multiply together to get a polynomial. It breaks down a complex expression into simpler factors. Think of it as reversing the process of expanding expressions.

When dealing with polynomials, look for common factors like numbers or variables that appear in each term. Sometimes, polynomials can include binomials or trinomials. Let’s dive deeper:

• Find Common Factors: Check each term to see if there’s a number or variable you can factor out. For example, in the polynomial \(9x^2 + 18x\), both terms have a common factor of 9x.
• Factor by Grouping: Group terms that have a common factor and factor them out. This technique is useful for more complicated polynomials.
• Prime Factorization: If each term is a perfect square or cube, it might be broken down further. For instance, square terms can be decomposed into a pair of identical factors.

Factoring helps us simplify the expression, making it easier to work with when performing operations like determining the LCM.

Simplifying Polynomials

When we simplify polynomials, we're making them easier to understand and work with. Simplification involves combining like terms and making an expression as concise as possible.

For example, the polynomial \(9x(x+2)\) becomes \(9x^2 + 18x\) when simplified. Here’s how you might tackle that:

• Distribute and Combine: Distribute any coefficients and combine like terms, which are terms that have the same variable raised to the same power.
• Expand Expressions: If the expression has squared binomials or similar, multiply them out. This helps achieve a simpler form.
• Check for Constants and Single Terms: Ensure all terms have been simplified to their simplest form, looking for constants or single terms that can be easily simplified.

Simplification is an important step before moving on to more complex operations involving polynomials, such as determining common multiples.

Least Common Multiple

The least common multiple (LCM) of polynomials is the smallest polynomial that each of the given polynomials divides. This term refers to finding a common ground between different expressions.

To find the LCM, consider the factors of each polynomial:

• Prime Factors: List the factors of each polynomial, identifying prime factors and powers of variables.
• Highest Power: Take each factor at its highest power as it appears in either polynomial. This ensures that the LCM is divisible by both original polynomials.
• Combine All Distinguished Terms: Multiply these factors together to find the LCM. For instance, for polynomial LCMs, incorporate coefficients and variable parts with their highest exponents.

As shown in the solution, the LCM of the given polynomials is \(36x^2(x+2)^2\). This process of finding the LCM is fundamental for solving equations involving multiple polynomial expressions.