Problem 66 - Algebra A Combined Function

Question

Factor. $$ 8 a b^{3}+27 a^{4} $$

Step-by-Step Solution

Answer

The factored form is \(a(2b + 3a)(4b^2 - 6ab + 9a^2)\).

Step 1: Identify a common factor

First, look for any common factors in the terms of the expression \(8ab^{3} + 27a^{4}\). Notice that both terms have a common factor \(a\).

Step 2: Factor out the common factor

Factor out \(a\) from the expression:\[8ab^3 + 27a^4 = a(8b^3 + 27a^3)\]

Step 3: Recognize the structure of the remaining terms

The remaining expression inside the parentheses is \(8b^3 + 27a^3\). Notice that this is a sum of cubes. Recall that a sum of cubes \(x^3 + y^3\) can be factored using the formula:\[x^3 + y^3 = (x + y)(x^2 - xy + y^2)\]

Step 4: Apply the sum of cubes formula

In \(8b^3 + 27a^3\), identify \(x = 2b\) and \(y = 3a\). Substitute into the sum of cubes formula:\[(2b + 3a)((2b)^2 - (2b)(3a) + (3a)^2)\] Simplifying each term:\[(2b + 3a)(4b^2 - 6ab + 9a^2)\]

Step 5: Write the complete factored form

Combine the factored common term from Step 2 with the factored expression from Step 4:\[a(2b + 3a)(4b^2 - 6ab + 9a^2)\] This is the fully factored form of the initial expression.

Key Concepts

Common Factor

Finding a common factor is a crucial first step in factoring polynomials. In the expression given, the terms are 8ab^3 and 27a^4. To identify a common factor, examine the components of each term. Here, both terms include the variable 'a'. This implies 'a' is a common factor.
An efficient way to spot common factors is to break each term down into its prime factors or use the greatest common factor (GCF) technique. In this exercise, factoring out a common factor 'a' simplifies the expression to:

• \( 8ab^3 + 27a^4 = a(8b^3 + 27a^3) \)

This process is analogous to pulling out a shared element, making the polynomial easier to handle in subsequent steps.

Sum of Cubes

After factoring out the common factor, the next focus is the expression within the parentheses, 8b^3 + 27a^3. This represents a classic case of the sum of cubes.
A sum of cubes can be expressed generally as \( x^3 + y^3 \) and can be decomposed using a specific algebraic identity:

• \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \)

In the expression 8b^3 + 27a^3, recognize that 8b^3 is \( (2b)^3 \) and 27a^3 is \( (3a)^3 \). By substituting these into the formula:

• \( (2b + 3a)((2b)^2 - (2b)(3a) + (3a)^2) = (2b + 3a)(4b^2 - 6ab + 9a^2) \)

This transformation simplifies the polynomial further by exploiting the algebraic structure inherent in cubes.

Algebraic Identities

Algebraic identities like the sum of cubes are powerful tools in simplifying and factoring polynomials. They enable the decomposition of complex expressions into more manageable forms using well-established formulas.
The identity \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \) operates as a blueprint for factoring expressions that conform to this pattern. By identifying components like \( x = 2b \) and \( y = 3a \), polynomials such as 8b^3 + 27a^3 can be systematically dismantled and expressed in a product form.
Understanding these identities not only aids in solving specific problems more easily but also enhances overall algebraic fluency, allowing you to recognize patterns and connections across different types of polynomial equations.