Finding the LCM of Polynomials

To find the Least Common Multiple (LCM) of polynomials, we first need to factor each polynomial completely into its prime factors. Then, we take each unique prime factor raised to the highest power it appears in any of the factorizations.

Step 1: Factor the first polynomial, \(x^3 + 8\)

The first polynomial is \(x^3 + 8\). This is a sum of cubes, which factors using the formula \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 2\).

So, the factored form of \(x^3 + 8\) is \((x+2)(x^2 - 2x + 4)\).

Step 2: Factor the second polynomial, \(x^2 + 5x + 6\)

The second polynomial is \(x^2 + 5x + 6\). This is a quadratic trinomial. We look for two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3.

So, the factored form of \(x^2 + 5x + 6\) is \((x+2)(x+3)\).

Step 3: Factor the third polynomial, \(x^3 + 4x^2 + 4x\)

The third polynomial is \(x^3 + 4x^2 + 4x\). First, we can factor out the common factor \(x\).

The expression inside the parenthesis, \(x^2 + 4x + 4\), is a perfect square trinomial, which factors using the formula \(a^2 + 2ab + b^2 = (a+b)^2\). Here, \(a = x\) and \(b = 2\).

So, the factored form of \(x^3 + 4x^2 + 4x\) is \(x(x+2)^2\).

Step 4: Identify the prime factors and their highest powers

Let's list the factored forms of the three polynomials:

• \(x^3 + 8 = (x+2)(x^2 - 2x + 4)\)
• \(x^2 + 5x + 6 = (x+2)(x+3)\)
• \(x^3 + 4x^2 + 4x = x(x+2)^2\)

Now, we identify all the unique prime factors across these factorizations and note their highest power:

• The factor \(x\) appears in \(x(x+2)^2\) with a power of 1. The highest power is \(x^1\).
• The factor \((x+2)\) appears in \((x+2)(x^2 - 2x + 4)\) with power 1, in \((x+2)(x+3)\) with power 1, and in \(x(x+2)^2\) with power 2. The highest power is \((x+2)^2\).
• The factor \((x+3)\) appears in \((x+2)(x+3)\) with power 1. The highest power is \((x+3)^1\).
• The factor \((x^2 - 2x + 4)\) appears in \((x+2)(x^2 - 2x + 4)\) with power 1. The highest power is \((x^2 - 2x + 4)^1\).

Step 5: Calculate the LCM

The LCM is the product of these unique factors raised to their highest powers:

Step 6: Compare with the options

Comparing our calculated LCM with the given options:

• Option 1: \(x (x + 2)² (x + 3) (x² – 2x + 4)\)
• Option 2: \(x (x – 2)² (x – 3) (x² + 2x + 4)\)
• Option 3: \((x + 2)² (x + 3) (x² – 2x + 4)\)
• Option 4: \((x – 2)² (x – 3) (x² – 2x + 4)\)

Our calculated LCM, \(x(x+2)^2(x+3)(x^2 - 2x + 4)\), matches Option 1.

Revision Table: Factoring Polynomials

Additional Information: LCM and GCF of Polynomials

Understanding LCM for polynomials is similar to finding the LCM for numbers. The Least Common Multiple is the smallest polynomial that is a multiple of all the given polynomials.

Another related concept is the Greatest Common Factor (GCF) of polynomials. To find the GCF, you take the common prime factors raised to the lowest power they appear in any of the factorizations.

For the polynomials \(x^3 + 8\), \(x^2 + 5x + 6\), and \(x^3 + 4x^2 + 4x\):

• Factored forms: \((x+2)(x^2-2x+4)\), \((x+2)(x+3)\), \(x(x+2)^2\)
• The only common factor is \((x+2)\).
• The lowest power of \((x+2)\) among the factorizations is \( (x+2)^1 \).

Therefore, the GCF of these three polynomials is \((x+2)\).

The relationship between LCM and GCF for two polynomials \(P(x)\) and \(Q(x)\) is \(P(x) \cdot Q(x) = \text{LCM}(P(x), Q(x)) \cdot \text{GCF}(P(x), Q(x))\). However, this relationship does not directly extend to three or more polynomials in the same simple form.