Problem 54 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$125 x^{3}+27 y^{3}$$

Step-by-Step Solution

Answer

The expression factors to \((5x + 3y)(25x^2 - 15xy + 9y^2)\).

Step 1: Identify the form

The expression given is \(125x^3 + 27y^3\). This follows the sum of two cubes pattern \(a^3 + b^3\), where \(a = 5x\) and \(b = 3y\). This identification is based on \((5x)^3 = 125x^3\) and \((3y)^3 = 27y^3\).

Step 2: Write the sum of two cubes formula

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). We will use this formula to factor the expression.

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

Substitute \(a = 5x\) and \(b = 3y\) into the sum of cubes formula: \[ (5x)^3 + (3y)^3 = (5x + 3y)((5x)^2 - 5x \cdot 3y + (3y)^2) \]

Step 4: Simplify each term in the expression

Compute the terms:- \((5x)^2 = 25x^2\)- \(- 5x \cdot 3y = - 15xy\)- \((3y)^2 = 9y^2\)So, the expression becomes:\[ (5x + 3y)(25x^2 - 15xy + 9y^2) \]

Step 5: Write the final factored form

The expression \(125x^3 + 27y^3\) is factored as \((5x + 3y)(25x^2 - 15xy + 9y^2)\).

Key Concepts

Understanding the Sum of Cubes

When dealing with polynomial expressions, you may encounter terms that are cubes of numbers or variables. The sum of cubes occurs when you have an expression of the form \(a^3 + b^3\). This kind of expression can be rewritten using a special formula. For example:

• If you identify that you have \(a^3 + b^3\), you can factor it as \((a + b)(a^2 - ab + b^2)\).
• This pattern is particularly useful because it simplifies what might initially seem like a complex polynomial into a product of two simpler expressions.
• In our original exercise, the expression \(125x^3 + 27y^3\) was identified as following the sum of cubes pattern.

Recognizing the sum of cubes allows you to factor polynomials effectively. First, identify the terms as perfect cubes, then apply the formula. This process simplifies solving equations further down the line.

Exploring the Difference of Cubes

Similar to the sum of cubes, the difference of cubes is another expression pattern you might encounter in polynomial factoring. This pattern takes the form \(a^3 - b^3\) and can be factored using a specific formula as well:

• The formula for the difference of cubes is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
• While the original exercise didn't involve a difference of cubes, understanding this pattern is equally crucial, as it helps factor polynomials that subtract one cube from another.
• Identify the cubes, then directly apply the formula to transform the expression into a product of smaller terms.

Mastering both sum and difference of cubes ensures you're well-prepared to tackle numerous polynomial expressions, helping to simplify complex algebraic equations.

Polynomial Expressions Simplified

Polynomial expressions can often feel complex, but understanding how to factor them can make a huge difference. These expressions consist of terms joined by plus or minus signs and can be simplified significantly through factoring.

• In the case of cubes, factoring involves recognizing specific patterns—like sum and difference of cubes—and applying the appropriate formulas.
• Being able to identify terms like \(a^3 + b^3\) or \(a^3 - b^3\) as known patterns makes the factoring process straightforward.
• Factoring polynomials is not just about simplifying expressions; it also prepares them for further operations, such as solving equations or integrating functions.

Breaking down complex polynomial expressions into simpler parts is a key skill in algebra. By practicing the recognition of patterns like the sum and difference of cubes, you'll enhance your mathematical toolkit.