Problem 49 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$27 x^{3}+64 y^{3}$$

Step-by-Step Solution

Answer

Factor the expression as \((3x + 4y)(9x^2 - 12xy + 16y^2)\).

Step 1: Recognize the Pattern

The expression given is in the form \( a^3 + b^3 \), which is a sum of cubes. We can use the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).

Step 2: Identify the Cubes

We identify \( a^3 = 27x^3 \) and \( b^3 = 64y^3 \). This gives us \( a = (27x^3)^{1/3} = 3x \) and \( b = (64y^3)^{1/3} = 4y \).

Step 3: Apply the Formula

Apply the sum of cubes formula. Substitute \( a = 3x \) and \( b = 4y \) into the formula: \( (3x + 4y)((3x)^2 - 3x \cdot 4y + (4y)^2) \).

Step 4: Calculate the Inner Expression

Calculate the squared and product terms: \( (3x)^2 = 9x^2 \), \(-3x \cdot 4y = -12xy \), \((4y)^2 = 16y^2 \).

Step 5: Write the Final Factored Form

Combine these results into the expression: \( (3x + 4y)(9x^2 - 12xy + 16y^2) \).

Key Concepts

Sum of Cubes

When we talk about the "sum of cubes," we are diving into an interesting algebraic identity. This identity helps us break down expressions where two cube terms are added together. For example, the expression \( a^3 + b^3 \) follows the sum of cubes pattern.

The formula for factoring a sum of cubes is:

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

The process itself is straightforward, but requires careful identification of the cube roots. Once identified, use the formula to factor the expression easily. In our example, knowing that \( 27x^3 + 64y^3 \) is a sum of cubes lets us utilize this helpful formula to get order from the chaos. It streamlines handling complex polynomial expressions.

Difference of Cubes

"Difference of cubes" is another powerful tool in your algebra toolkit. It applies when you need to factor expressions that subtract one cube from another. Thus, for any expression like \( a^3 - b^3 \), knowing this pattern is crucial.

The formula for factoring a difference of cubes goes like this:

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

This process involves identifying the cube roots of each term, much like the sum of cubes. It’s not just reversing signs; it’s about keeping the parts in correct positions.

Even though our original exercise focused on the sum of cubes, the "difference of cubes" is equally important and appears frequently in polynomial expressions.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving multiple terms. These can be constants, variables, or both raised to different powers. They appear frequently in algebra and are fundamental to understanding how to manipulate and solve equations.

For instance, in a polynomial like \( 27x^3 + 64y^3 \), we have two cube terms. Factoring such expressions sometimes requires identifying specific patterns, like sums or differences of cubes, to simplify them.

Learning to identify patterns in polynomial expressions can significantly ease the factoring process. These skills help in simplifying complex algebraic problems into manageable parts.

Algebraic Identities

Algebraic identities are pre-defined equations that hold true for all variable values. They form the core of algebra, allowing us to simplify and reorganize expressions efficiently.

Some common algebraic identities include:

• Sum of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
• Difference of cubes: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

These identities save time because they eliminate the need for lengthy calculations, offering ready-made solutions.

By understanding and utilizing these identities, students can instantly recognize parts of polynomial expressions and factor them with ease in their studies.