Problem 52 - Algebra 2

Question

Find the LCM of each set of polynomials. \(a^{2}-2 a-3, a^{2}-a-6\)

Step-by-Step Solution

Answer

The LCM is \((a - 3)(a + 1)(a + 2)\) or \(a^3 - 7a - 6\).

Step 1: Factor Each Polynomial

First, factor each of the given polynomials completely. For the polynomial \(a^2 - 2a - 3\), we look for two numbers that multiply to \(-3\) and add up to \(-2\). These are \(-3\) and \(1\), so \(a^2 - 2a - 3 = (a - 3)(a + 1)\). For the polynomial \(a^2 - a - 6\), we need two numbers that multiply to \(-6\) and add up to \(-1\), which are \(-3\) and \(2\), thus \(a^2 - a - 6 = (a - 3)(a + 2)\).

Step 2: Identify All Unique Factors

List all unique factors obtained from the factored forms of both polynomials. From \((a - 3)(a + 1)\) and \((a - 3)(a + 2)\), the unique factors are \(a - 3\), \(a + 1\), and \(a + 2\).

Step 3: Construct the LCM

The least common multiple (LCM) is the product of the highest powers of all unique factors. Here, each factor \(a - 3\), \(a + 1\), and \(a + 2\) appears once without any repeated powers, leading to the LCM being \((a - 3)(a + 1)(a + 2)\).

Step 4: Expand the LCM if Necessary

If required, expand the expression \((a - 3)(a + 1)(a + 2)\) using the distributive property. First, expand \((a + 1)(a + 2) = a^2 + 3a + 2\). Then multiply by \(a - 3\): \((a^2 + 3a + 2)(a - 3) = a^3 - 3a^2 + 3a^2 - 9a + 2a - 6 = a^3 - 7a - 6\).

Step 5: Conclusion

The LCM of the polynomials \(a^2 - 2a - 3\) and \(a^2 - a - 6\) is \((a - 3)(a + 1)(a + 2)\) or, when expanded, \(a^3 - 7a - 6\).

Key Concepts

Factoring Polynomials

Factoring polynomials is like unraveling a stitched piece of fabric into its original threads. It's a crucial step in many algebraic processes, including finding the least common multiple (LCM). When we factor polynomials, we look for simpler expressions that multiply together to give us the original polynomial.
To factor a quadratic polynomial like the ones in this exercise, you can follow these steps:

• Identify two numbers that multiply to the constant term (the number without a variable) and add up to the middle coefficient (the number in front of the variable with an exponent of 1).
• Use these numbers to break down the middle term and group the polynomial into factorable pairs.
• Factor out the common factors from each group to find the factored form.

For example, for the polynomial \(a^2 - 2a - 3\), the numbers \(-3\) and \(1\) fit because they multiply to \(-3\) and add to \(-2\). Thus, we factor it as \((a - 3)(a + 1)\). Factoring lays the foundation for identifying unique factors in the next step.

Unique Factors

When working with polynomials and looking to find their least common multiple, identifying unique factors is essential. Unique factors are the individual expressions that appear in the factored form of the polynomials being considered.
These are important because the LCM must include every factor that appears, at least once, in any of the expressions.

• Start by factoring each polynomial completely, as was done in the previous section.
• Once factored, list out all different factors. A unique factor is one that appears at least once in the factorization of any of the polynomials.

For the polynomials \((a - 3)(a + 1)\) and \((a - 3)(a + 2)\), our unique factors become \(a - 3\), \(a + 1\), and \(a + 2\). These unique factors ensure you have accounted for all necessary parts of each polynomial to create the LCM. Once you identify these unique pieces, you're ready to construct your LCM.

Expanding Expressions

Expanding expressions is the final step that makes the multiplication of expressions more explicit, revealing all the terms. It's often done after finding the LCM to express the solution in a single polynomial form.
Expansion uses the distributive property to remove parentheses and show each term of the final expression in its entirety.

• Begin by expanding binomials one pair at a time, especially if you have more than two factors.
• Use the distributive method: distribute each term of one factor across the terms of another factor.
• Combine like terms to form a single simplified expression without parentheses.

In this problem, \((a - 3)(a + 1)(a + 2)\) expands in stages. First expand \((a + 1)(a + 2)\) to get \(a^2 + 3a + 2\). Then, multiply this result by \(a - 3\) and carefully apply the distributive property: \(a^3 - 3a^2 + 3a^2 - 9a + 2a - 6\), which simplifies to \(a^3 - 7a - 6\). Expanding solidifies your understanding of all components in the final polynomial result.