Problem 60 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ x^{6}+64 $$

Step-by-Step Solution

Answer

The factorization is \( (x^2 + 4)(x^4 - 4x^2 + 16) \).

Step 1: Recognize the special expression

The expression \( x^6 + 64 \) is a sum of cubes, which can be factored using the sum of cubes formula: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). We need to express the terms in cube form.

Step 2: Express terms as cubes

Here, \( x^6 = (x^2)^3 \) and \( 64 = 4^3 \). So, the expression becomes \( (x^2)^3 + 4^3 \). This fits the form \( a^3 + b^3 \) where \( a = x^2 \) and \( b = 4 \).

Step 3: Apply the sum of cubes formula

Substitute \( a = x^2 \) and \( b = 4 \) into the sum of cubes formula: \( (a + b)(a^2 - ab + b^2) \). This gives \( (x^2 + 4)((x^2)^2 - x^2 \cdot 4 + 4^2) \).

Step 4: Simplify the factored expression

Simplify the expression \( (x^2)^2 - x^2 \cdot 4 + 4^2 \) to \( x^4 - 4x^2 + 16 \). The factored form is \( (x^2 + 4)(x^4 - 4x^2 + 16) \).

Key Concepts

Sum of Cubes

The sum of cubes is a common algebraic expression that comes into play when dealing with polynomial factoring. Recognizing when an expression is a sum of cubes helps simplify complex polynomials. In mathematics, the sum of cubes formula is expressed as:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

Breaking down this formula:

• \(a + b\) is the sum of the cube roots.
• \(a^2 - ab + b^2\) is the trinomial that perfectly pairs with the sum of cube roots to complete the factorization.

This formula can sometimes be tricky to spot, but once identified, it significantly helps simplify complex polynomials, just like in our problem where \(x^6 + 64\) was rewritten as \((x^2)^3 + 4^3\). This fit the general \(a^3 + b^3\) form, allowing us to apply the formula and turn it into a product of simpler expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables (like \(x\) or \(y\)), and operations (like addition or multiplication). These expressions can take many different forms and are foundational in algebra for representing real-world scenarios or mathematical concepts. For example:

• \(3x + 5\)
• \(2x^2 - y + 6\)

To solve algebraic expressions, one often needs to simplify or factor them. Simplifying involves combining like terms or executing mathematical operations. Factoring involves expressing the expression as a product of its factors, which is pivotal when solving equations. In the context of \(x^6 + 64\), identifying it as a sum of cubes allows for factoring, which simplifies the polynomial into more manageable parts.

Polynomial Factoring

Factoring is a critical skill in algebra that involves breaking down polynomials into products of simpler polynomials. Polynomials are sums of terms that might have coefficients and variables raised to various powers. Let's break down why this is important:

• Factoring simplifies equations, making them easier to solve.
• It helps identify roots and understand the behavior of functions.
• Factoring allows for simplification or cancellation in expressions, particularly useful in calculus and algebraic manipulations.

In the exercise with \(x^6 + 64\), recognizing this as a sum of cubes allowed the application of a specific formula. This direct factorization gave \((x^2 + 4)(x^4 - 4x^2 + 16)\). Breaking down these expressions into smaller parts through factoring provides a clearer view, makes solving equations relating to them more straightforward, and helps in graphing their functions.