Problem 50 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$8 x^{3}+27 y^{3}$$

Step-by-Step Solution

Answer

The expression is factored as \((2x + 3y)(4x^2 - 6xy + 9y^2)\).

Step 1: Identifying the Formula

Recognize that we need to use the sum of cubes formula, which is: \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \).

Step 2: Identifying the Terms

The given expression is \(8x^3 + 27y^3\). Here, identify \(a = 2x\) and \(b = 3y\), since \((2x)^3 = 8x^3\) and \((3y)^3 = 27y^3\).

Step 3: Applying the Formula

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

Step 4: Simplifying the Expression

Now simplify the expression: \((2x + 3y)(4x^2 - 6xy + 9y^2)\).

Step 5: Final Factored Form

Thus, the factored form of the expression \(8x^3 + 27y^3\) is \((2x + 3y)(4x^2 - 6xy + 9y^2)\).

Key Concepts

Sum of Cubes

The sum of cubes refers to expressions that take the form \( a^3 + b^3 \). It can be factored using a special algebraic identity. The sum of cubes formula is: \[a^3 + b^3 = (a+b)(a^2-ab+b^2)\]This formula helps break down the polynomial into simpler binomial and trinomial expressions.

To apply this formula:

• Identify the cube root of each term. For example, if you have \(8x^3 + 27y^3\), the cubes are \(2x\) and \(3y\).
• Substitute these values into the formula: \((2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)\).
• Simplify the resulting expression to obtain the factored form \((2x + 3y)(4x^2 - 6xy + 9y^2)\).

This method provides a straightforward way to factor sums of cubes, making polynomial factoring manageable and systematic.

Difference of Cubes

The difference of cubes follows a similar logic as the sum of cubes but addresses expressions that look like \( a^3 - b^3 \). There is also an established formula for factoring these expressions: \[a^3 - b^3 = (a-b)(a^2+ab+b^2)\]This identity helps decompose the polynomial into a product of simpler expressions.

Here's how you can use it:

• Determine the cube root of each component. If given a polynomial like \(27x^3 - 8y^3\), the cube roots are \(3x\) and \(2y\).
• Substitute these into the formula \((3x - 2y)((3x)^2 + (3x)(2y) + (2y)^2)\).
• Simplify the expression to get it fully factored, resulting in \((3x - 2y)(9x^2 + 6xy + 4y^2)\).

This approach can be a lifesaver for tackling expressions like \( a^3 - b^3 \), allowing for a quicker route to polynomial simplification.

Algebraic Identities

Algebraic identities are powerful tools in algebra. They are formulas that provide universally applicable shortcuts for simplifying and factoring mathematical expressions. The identities for sum and difference of cubes are perfect examples.

Some of the key benefits of these identities include:

• They offer a quick route to factor otherwise complicated polynomials.
• They reduce complex algebraic problems into simpler, more manageable expressions.
• They are universally applicable, saving time across a variety of problems.

Understanding these identities and how to apply them can transform your approach to algebra. It allows you to see complicated expressions as a series of manageable parts, each one fitting perfectly into the other, just like pieces in a puzzle.