Problem 44 - Elementary and Intermediate Algebra

Question

Factor. $$ 2 x^{3}+54 $$

Step-by-Step Solution

Answer

The expression factors to \(2(x + 3)(x^2 - 3x + 9)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look for the greatest common factor of the terms in the expression. Both terms, \(2x^3\) and \(54\), are divisible by 2. So, the GCF is 2.

Step 2: Factor Out the GCF

Divide each term in the expression by the GCF, which is 2. This gives us: \(2(x^3 + 27)\).

Step 3: Recognize a Sum of Cubes

The expression inside the parentheses, \(x^3 + 27\), can be recognized as a sum of cubes because \(x^3 = (x)^3\) and \(27 = 3^3\).

Step 4: Apply the Sum of Cubes Formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 3\). Substitute these values into the formula to get \((x + 3)(x^2 - 3x + 9)\).

Step 5: Write the Final Factored Form

Combine the GCF we factored out earlier with the factored form of the sum of cubes: \(2(x + 3)(x^2 - 3x + 9)\). This is the fully factored expression of the original polynomial.

Key Concepts

Greatest Common Factor

When factoring expressions, the first step is usually to identify the Greatest Common Factor (GCF). The GCF is the largest number or variable that evenly divides each term in an expression. It simplifies the expression and prepares it for further factoring.

To find the GCF, begin by listing the factors of each term. For the expression \(2x^3 + 54\), examine both terms: \(2x^3\) and \(54\).

• Factors of \(2x^3\) include: 1, 2, \(x\), \(x^2\), \(x^3\)
• Factors of 54 include: 1, 2, 3, 6, 9, 18, 27, 54

The common factor present in both lists is 2.

Thus, the GCF of the entire expression is 2. By factoring out 2, you simplify the expression, which helps in identifying other factorizable structures, like sums of cubes.

Sum of Cubes

The expression \(x^3 + 27\) within the parentheses \(2(x^3 + 27)\) is a classic example of a sum of cubes. The sum of cubes refers to the addition of two cube terms. Recognizing these expressions is vital for applying further factoring techniques.

A sum of cubes follows the pattern \(a^3 + b^3\), where the terms can be rewritten as cubes of two other terms. Here, \(x^3\) is \((x)^3\) and 27 is \((3)^3\). Once you identify a sum of cubes, you can apply the sum of cubes formula:

• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Substitute \(a = x\) and \(b = 3\) into the formula:

• \((x + 3)((x)^2 - (x)(3) + (3)^2) = (x + 3)(x^2 - 3x + 9)\)

This structured approach reveals the fully factored form of the sum of cubes inside the expression.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler polynomials that, when multiplied together, will give you the original polynomial. It often combines several factoring strategies, including identifying the Greatest Common Factor (GCF) and recognizing special forms like the sum of cubes.

To factor the polynomial \(2x^3 + 54\) completely:

• First, factor out the GCF: \(2\).
• This leaves you with \(2(x^3 + 27)\).
• Next, identify the remaining expression \(x^3 + 27\) as a sum of cubes.
• Apply the sum of cubes formula to further factor \(x^3 + 27\) to \((x + 3)(x^2 - 3x + 9)\).

Combine these steps to write the fully factored expression: \(2(x + 3)(x^2 - 3x + 9)\).

Utilizing these organized steps simplifies the entire factoring process, allowing you to handle complex polynomials with confidence.