Problem 46 - Elementary Algebra

Question

Factor completely. $$ (x-4) 3-27 $$

Step-by-Step Solution

Answer

The complete factorization is \((x-7)(x^2 - 5x + 13)\).

Step 1: Simplify Expression

First, simplify the given expression: $$(x-4)^3 - 27.$$ Notice that the expression is a difference of cubes. The formula for the difference of cubes is: $$a^3 - b^3 = (a-b)(a^2 + ab + b^2).$$

Step 2: Identify Cubes

Identify the cubes in the expression $(x-4)^3 - 27$. Here, we see that $$a = (x-4)$$ and $$b = 3$$ (since $3^3 = 27$).

Step 3: Apply Difference of Cubes Formula

Apply the difference of cubes formula: $$(x-4)^3 - 3^3 = ((x-4) - 3)((x-4)^2 + (x-4)(3) + 3^2).$$

Step 4: Simplify the Expression

Simplify each part of the factorization: 1. $((x-4) - 3) = (x - 7)$. 2. $((x-4)^2 + 3(x-4) + 3^2)$ becomes: a. $(x-4)^2 = x^2 - 8x + 16$ b. $3(x-4) = 3x - 12$ c. $3^2 = 9$ Combine: $x^2 - 8x + 16 + 3x - 12 + 9$. Combine terms: $$x^2 - 5x + 13.$$

Step 5: Write Final Factored Form

Combine the simplified factors: So the complete factorization is: $$(x-7)(x^2 - 5x + 13).$$

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental process in algebra where you express a mathematical expression as a product of its factors. The factors are simpler expressions that, when multiplied together, recreate the original expression. This technique is particularly helpful for simplifying expressions and solving equations.

For example, if you have an expression such as \((x - 4)^3 - 27\), you can factor it into a simpler form using special formulas like the difference of cubes. Factoring helps by breaking down complex expressions, making it easier to work with them and solve for unknown variables if needed.

Ultimately, the skill in factoring involves recognizing patterns and choosing the correct method to break down the expression into its simplest parts. This ultimately aids in simplifying and solving equations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form, which makes them much more manageable to work with. This typically involves combining like terms and reducing any coefficients or terms that appear in the expression.

In our example, after factoring \((x - 4)^3 - 27\) using the difference of cubes formula, we are left with the expression: \((x - 7)(x^2 - 5x + 13)\). To simplify further, we can break down any complex components, such as squaring \((x - 4)\) and performing multiplication that occurs in the process.

Once you have broken down these components into smaller, more manageable parts, combining them will yield a cleaner, more straightforward result. In this scenario, the expression was already simplified after factorization, resulting in: \((x - 7)(x^2 - 5x + 13)\). Simplifying helps in revealing the underlying structure of algebraic expressions and makes future calculations and solutions more straightforward.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. It is an essential part of algebra and often involves using formulas or techniques to simplify the polynomial to its component factors.

In our given expression \((x - 4)^3 - 27\), we identify it as a difference of cubes. Recognizing this pattern allows us to use a special factoring formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

After applying this formula, the expression transforms into: \((x - 7)(x^2 - 5x + 13)\). This factorization illustrates how polynomial expressions can often be rewritten as a product of simpler binomials or trinomials, making them easier to handle, especially in solving for variable values.

Polynomial factorization is not only about simplifying expressions but also about understanding the structure and relationship between different polynomial components, which is crucial in more advanced algebra and calculus concepts.