Question

Factor the sum or difference of two cubes. $$ 8 x^{3}+27 $$

Short answer

The factored form is (2x + 3)(4x^{2} - 6x + 9).

Step-by-step solution

Identify the equation form

Recognize that the given expression is a sum of cubes, which can be written as: a^{3} + b^{3} or a^{3} - b^{3}. In this case, the expression is a sum of cubes: 8x^{3} + 27.

Rewrite each term as a cube

Rewrite each term in the form of something cubed: 8x^{3} = (2x)^{3} 27 = 3^{3}. Thus, the expression becomes: (2x)^{3} + 3^{3}.

Apply the sum of cubes formula

Use the sum of cubes formula: a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}). Here, a = 2x and b = 3. So substitute these values into the formula:

Perform the substitution and simplify

Substitute a = 2x and b = 3 into the sum of cubes formula: (2x + 3)((2x)^{2} - (2x)(3) + 3^{2}). Simplify within the parentheses: (2x + 3)(4x^{2} - 6x + 9).

Write the final answer

The factored form of the original expression 8x^{3} + 27 is (2x + 3)(4x^{2} - 6x + 9).

Key concepts

Factoring Polynomials

Factoring polynomials is a fundamental concept in algebra that involves breaking down a polynomial into simpler elements called factors. This process helps in solving equations more easily and understanding the behavior of the polynomial. In the given exercise, we dealt with the sum of cubes, which is a specific type of polynomial.

For example, the expression \(8x^{3} + 27\) is a polynomial that can be factored using the sum of cubes formula. Factoring such expressions involves a few systematic steps:

• Identify the form of the polynomial.
• Rewrite each term as a cube.
• Apply the appropriate polynomial identity formula.
• Simplify the expression to its factored form.

Understanding these steps is crucial for mastering polynomial factoring. Let's dive into more details about each step.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and arithmetic operations. They do not have an equality sign, in contrast to algebraic equations which do. For instance, in the exercise \(8x^{3} + 27\) is an algebraic expression.

Understanding how to manipulate algebraic expressions is essential for solving various algebra problems. Here are some key points about algebraic expressions:

• Variables: Symbols representing unknown values, such as \(x\).
• Constants: Fixed values such as numbers (e.g., 27).
• Operations: Basic arithmetic like addition (+), subtraction (-), multiplication (×), and division (÷).

By mastering these components, students can better navigate problems like factoring sums and differences of cubes. Rewriting each term in proper forms, like cubes, simplifies the factorization process better.

Polynomial Identities

Polynomial identities are standard formulas that represent how polynomials can be factored. These identities are mathematical truths that hold for all values of the variables involved. One such identity applied in the exercise is the sum of cubes formula.

For the sum of cubes, the formula is: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Here, \(a\) and \(b\) represent any expressions. Using this identity helps in breaking down complex polynomials into simpler multiplicative factors.

Let's see the steps the exercise showcases:

• Recognize the form: Identify the polynomial as a sum of cubes.
• Rewrite terms: Express each term as a cube (e.g., \(8x^{3} = (2x)^{3}\) and \(27 = 3^{3}\)).
• Apply the identity: Use the sum of cubes formula to factor the expression.
• Simplify: Perform algebraic operations to simplify within parentheses.

This structured approach simplifies the polynomial and aids in solving algebraic equations effectively.