Problem 57 - College Algebra

Question

Factor using the formula for the sum or difference of tho cubes. $$ x^{3}+27 $$

Step-by-Step Solution

Answer

The factored form of \(x^{3}+27\) is \((x + 3)(x^{2} - 3x + 9)\).

Step 1: Identify A and B

To use our formula, we first need to identify A and B such that \(A^3 = x^3\) and \(B^3 = 27\). Here, A will be x and B will be 3 because \(x^3 = x^3\) and \(3^3 = 27\).

Step 2: Apply the formula

Substitute A and B into the formula, \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\), we get \(x^3 + 3^3 = (x + 3)(x^2 - 3x + 3^2)\).

Step 3: Simplify the equation

Next, simplify the equation: \(x^3 + 3^3 = (x + 3)(x^2 - 3x + 9)\). This cannot be simplified further and is the factored form of the expression.

Key Concepts

Sum and Difference of Cubes

When you're dealing with expressions like \(x^3 + 27\), it's helpful to recognize that these can sometimes be simplified using the sum and difference of cubes formula. This method helps to break down more complex expressions into manageable pieces. Understanding the sum and difference of cubes is crucial for factoring cubic polynomials efficiently.

For the sum of cubes, the formula is:

• \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\)

For the difference of cubes, the formula changes slightly to:

• \(A^3 - B^3 = (A - B)(A^2 + AB + B^2)\)

The key to using these formulas is identifying the values of \(A\) and \(B\) correctly. For example, in \(x^3 + 3^3\), we have \(A = x\) and \(B = 3\). Once these values are determined, you can substitute them into the appropriate formula. Recognizing these patterns can make factoring these expressions much simpler.

Factoring Expressions

Factoring expressions is a method used to simplify complex algebraic expressions. It involves breaking down an expression into a product of its factors. This process can significantly simplify solving algebraic equations. In the context of polynomials, it helps in finding roots more easily, among other applications.

For instance, given \(x^3 + 27\), once identified as a sum of cubes, you use the formula \((A + B)(A^2 - AB + B^2)\). This turns the expression into \((x + 3)(x^2 - 3x + 9)\), a product involving simpler factors. Always remember to

• Identify if an expression fits a specific pattern, like sum or difference of cubes.
• Accurately find the factors \(A\) and \(B\).
• Substitute into the proper formula, as outlined in patterns and identities.

Factoring not only aids in solving equations but also enhances understanding of the mathematical structure of expressions.

Algebraic Identities

Algebraic identities are mathematical truths that help simplify and solve equations more easily. They provide a quick and reliable way to deal with expressions by recognizing certain patterns. The identites for the sum and difference of cubes are critical examples of such patterns.

Using identities like \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\) streamlines the factoring process. It's important to be comfortable with these identities because they break complex equations down into multiple simpler factors, making computations easier.

When tackling any factoring problem:

• Identify the type of expression and relevant identity to use.
• Verify your results by multiplying the factors to ensure they expand back to the original expression.
• Practice regularly to recognize different algebraic patterns quickly and effectively.

Utilizing algebraic identities makes it possible to solve equations more efficiently, fostering a deeper understanding of mathematical principles.