Problem 81 - Elementary Algebra

Question

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ x^{3}+6 x^{2}+9 x ; x^{3}+7 x^{2}+12 x $$

Step-by-Step Solution

Answer

The LCM is \(x(x + 3)^2(x + 4)\).

Step 1: Factor each polynomial

Factor each polynomial using common factors and recognizable patterns. For the first polynomial, we have:\[x^3 + 6x^2 + 9x = x(x^2 + 6x + 9) = x(x + 3)^2\]For the second polynomial:\[x^3 + 7x^2 + 12x = x(x^2 + 7x + 12) = x(x + 3)(x + 4)\]

Step 2: Identify the unique factors

List the unique factors of each polynomial. From the factorizations:For \(x(x + 3)^2\), the factors are: \(x\) and \((x + 3)\).For \(x(x + 3)(x + 4)\), the factors are: \(x\), \((x + 3)\), and \((x + 4)\).

Step 3: Determine the highest power of each factor

Identify the highest power of each unique factor. Thus, the highest power of each factor is:\(x\): \(x\)\((x + 3)\): \((x + 3)^2\)\((x + 4)\): \((x + 4)\)

Step 4: Multiply the highest power of each factor

Multiply the highest powers of each identified factor to find the least common multiple. Therefore, the least common multiple (LCM) is:\[x(x + 3)^2(x + 4)\]

Key Concepts

Polynomial Factorization

To understand polynomial factorization, think of it like breaking down numbers into their prime factors. Here, we break down polynomials into simpler multiplied terms.

For example, take the polynomial given in the exercise:
\(x^3 + 6x^2 + 9x\).
By factoring out the greatest common factor (GCF), which is \(x\), you get:
\(x(x^2 + 6x + 9)\).
Next, notice that \(x^2 + 6x + 9\) is a perfect square trinomial and can be factored further:
\(x(x + 3)^2\).

For the second polynomial \(x^3 + 7x^2 + 12x\):
Again, pull out the GCF \(x\):
\(x(x^2 + 7x + 12)\).
This can be factored further into:
\(x(x + 3)(x + 4)\).

This process of breaking down each polynomial into simpler factors makes it easier to work with them in future steps, such as finding the least common multiple (LCM).

Highest Power of Factors

Identifying the highest power of each factor within the polynomials is crucial. Higher power means the larger exponent that appears for each factor.

Look at each factor we identified in the polynomials:
From \(x(x + 3)^2\), we have:

• \((x + 3)^2\)
• \(x\)

And from \(x(x + 3)(x + 4)\), we have:

• \(x\)
• \(x + 3\)
• \(x + 4\)

The highest power of each factor is:

• \(x\): Here, both polynomials have it as \(x^1\), so the highest power is \(x\).
• \(x + 3\): From \((x + 3)^2\) compared to \((x + 3)\), the highest power is \((x + 3)^2\).
• \(x + 4\): This factor appears only once as \((x + 4)\). Hence, \(x + 4\) is \(x + 4\).

Picking the highest powers ensures we capture all pieces needed for the LCM.

Unique Factors Identification

The next step in finding the LCM is identifying all unique factors. This means listing each distinct factor from both polynomials.

For the first polynomial, \(x(x + 3)^2\), we identified:

• \(x\)
• \(x + 3\)

For the second polynomial, \(x(x + 3)(x + 4)\), we have:

• \(x\)
• \(x + 3\)
• \(x + 4\)

Combining these, the unique factors are:

• \(x\)
• \(x + 3\)
• \(x + 4\)

Unique factors give us a complete set of elements we need to consider for the LCM. Ensuring no factor is missed is key to correctly determining the LCM.