Question

1. find the lcm of \\(x^2 + 5x + 6\\), \\(x^2 + 6x + 8\\), and \\(x^2 + 7x + 12\\)

\\((x+2)^2(x+3)(x+4)\\)
\\((x+2)(x+3)(x+4)\\)
\\((x+2)(x+3)(x+4)^2\\)
\\((x+2)(x+3)^2(x+4)\\)

Explanation:

🆕 New Concept Discovered: Least Common Multiple of Polynomials
Finding the shared expression of highest powers

Step 1: Factor the first polynomial

We factor the quadratic expression \(x^2 + 5x + 6\). We look for two numbers that multiply to \(6\) and add to \(5\). These numbers are \(2\) and \(3\):

$$x^2 + 5x + 6 = (x + 2)(x + 3)$$

Step 2: Factor the second polynomial

We factor the quadratic expression \(x^2 + 6x + 8\). We look for two numbers that multiply to \(8\) and add to \(6\). These numbers are \(2\) and \(4\):

$$x^2 + 6x + 8 = (x + 2)(x + 4)$$

Step 3: Factor the third polynomial

We factor the quadratic expression \(x^2 + 7x + 12\). We look for two numbers that multiply to \(12\) and add to \(7\). These numbers are \(3\) and \(4\):

$$x^2 + 7x + 12 = (x + 3)(x + 4)$$

Step 4: Find the Least Common Multiple (LCM)

To find the LCM of the factored polynomials, we take the highest power of each unique linear factor present across all three expressions:

• The factor \((x + 2)\) appears at most once in any single polynomial.
• The factor \((x + 3)\) appears at most once in any single polynomial.
• The factor \((x + 4)\) appears at most once in any single polynomial.

Combining these unique factors gives:

$$\text{LCM} = (x + 2)(x + 3)(x + 4)$$

Answer:

\((x + 2)(x + 3)(x + 4)\) (the second option)