Problem 49 - College Algebra

Question

In Exercises \(49-56,\) factor using the formula for the sum or difference of two cubes. $$x^{3}+27$$

Step-by-Step Solution

Answer

The factorized form of the equation \(x^3 + 27\) is \(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\)

Step 1: Identify the format of the expression

The given expression is \(x^3 + 27\). This can be seen as the sum of two cubes where \(a = x\) and \(b = 3\) since \(3^3 = 27\).

Step 2: Apply the sum of cubes formula

The formula for the sum of two cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Substituting \(a = x\) and \(b = 3\) into the formula, we get: \(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\).

Key Concepts

Sum of Cubes

The sum of cubes refers to the algebraic expression in the format of \[a^3 + b^3\]. Recognizing this pattern is key to factoring the expression correctly.
For the formula, we use:

• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) for the sum of cubes.

In our exercise, \(x^3 + 27\) can be seen as the sum of two cubes, where \(a = x\) and \(b = 3\), since \(3^3 = 27\).
This makes it straightforward to substitute into the formula. The result is a neatly factored polynomial: \[(x + 3)(x^2 - 3x + 9)\]. Working with sum of cubes can simplify solving complex polynomials.

Algebraic Expressions

Algebraic expressions combine numbers and variables using operations like addition, subtraction, multiplication, and division.
These expressions can often be rewritten in different forms to make solving equations or understanding relationships clearer.
Breaking down an algebraic expression involves recognizing its structure:

• Terms: Parts of an expression separated by addition or subtraction.
• Coefficients: Numerical factor of a term that contains a variable.
• Variables: Symbols representing numbers.
• Constants: Numbers without variables.

Understanding these components is essential for identifying patterns like the sum or difference of cubes.
It helps in choosing the correct formula or method to simplify and solve expressions, leading to clear and accurate solutions.

Polynomial Equations

Polynomial equations involve expressions where variables are raised to whole number powers and combined using operations. They are encountered frequently in algebra.
These equations can vary in complexity, but often require factoring for solutions.
The degree of the polynomial, given by the highest power of the variable, indicates the number of solutions.

• For example, a cubic polynomial (degree 3) like \(x^3 + 27\), may have up to 3 real solutions.

Factoring is a vital skill to solve polynomial equations, especially when they fit specific patterns, like the sum of cubes. Using the formula to break down \(x^3 + 27\) into \((x + 3)(x^2 - 3x + 9)\) is a step towards finding these solutions. Solving polynomial equations becomes more manageable with a systematic approach to factoring and simplifying.