Problem 64 - Algebra and Trigonometry

Question

Factor the expression completely. $$ x^{6}+64 $$

Step-by-Step Solution

Answer

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

Step 1: Recognize Sum of Two Terms

The given expression is \( x^6 + 64 \). Notice that this is a sum of two terms: \( x^6 \) and \( 64 \), which is a perfect sixth power. We can rewrite \( 64 \) as \( (2^3)^2 \) or simply \( 2^6 \). This expression is structured as \( a^6 + b^6 \), where \( a = x \) and \( b = 2 \).

Step 2: Apply the Sum of Two Even Powers Formula

To factor expressions like \( a^6 + b^6 \), we can use the identity: \( a^6 + b^6 = (a^2 + b^2)(a^4 - a^2b^2 + b^4) \). For our expression, we substitute \( a = x \) and \( b = 2 \):\[x^6 + 2^6 = (x^2 + 2^2)(x^4 - x^2 \, \cdot \, 2^2 + 2^4)\]

Step 3: Simplify Each Factor

Now, simplify the factors obtained in the previous step. For the first factor:\( x^2 + 2^2 = x^2 + 4 \).For the second factor:\( x^4 - x^2 \, \cdot \, 4 + 16 \), which simplifies to:\( x^4 - 4x^2 + 16 \).Thus, the complete factorization of \( x^6 + 64 \) is \((x^2 + 4)(x^4 - 4x^2 + 16)\).

Step 4: Verify the Factorization

To ensure the factorization is correct, we can expand \((x^2 + 4)(x^4 - 4x^2 + 16)\) to see if it results in \( x^6 + 64 \).First, distribute \( x^2 + 4 \) over \( x^4 - 4x^2 + 16 \):- \( x^2 \cdot x^4 = x^6 \)- \( x^2 \cdot (-4x^2) = -4x^4 \)- \( x^2 \cdot 16 = 16x^2 \)- \( 4 \cdot x^4 = 4x^4 \)- \( 4 \cdot (-4x^2) = -16x^2 \)- \( 4 \cdot 16 = 64 \)Combine like terms and you will get:\( x^6 + 0x^4 + 0x^2 + 64 \), which simplifies to \( x^6 + 64 \). This confirms that our factorization is correct.

Key Concepts

Sum of Even Powers

At first glance, factoring expressions like \( x^6 + 64 \) can be tricky, but understanding the concept of the 'sum of even powers' makes it a lot easier. When dealing with this type of expression, we identify each term as an even power. In this case:

• \( x^6 \) is already an even power.
• \( 64 \) can be rewritten as \( 2^6 \), another even power.

You can structure the expression as \( a^6 + b^6 \) with \( a = x \) and \( b = 2 \). Even powers have unique factorization properties that allow us to use specific algebraic identities to simplify them further. Recognizing this structure is the first step in applying the correct mathematical tools for factorization.

Algebraic Identities

Algebraic identities are essential tools for simplifying and factoring polynomial expressions. The sum of even powers utilizes a specific identity:\[ a^6 + b^6 = (a^2 + b^2)(a^4 - a^2b^2 + b^4)\]This identity helps in reducing the complexity of seemingly difficult expressions. Applying this to our example:

• Set \( a = x \) and \( b = 2 \).
• Substitute into the identity to get \((x^2 + 2^2)(x^4 - x^2 \cdot 2^2 + 2^4)\).

Once substituted, each part calculates to give a more manageable form:

• \( x^2 + 4 \)
• \( x^4 - 4x^2 + 16 \)

Use these algebraic identities in algebra to transform complex expressions into products of simpler factors.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Our task involves factoring such polynomial expressions. While the problem at hand is specific—factoring \( x^6 + 64 \)—it models the general principle of simplifying expressions and finding factorial components.

• First, recognize the overall structure of the polynomial.
• Identify any algebraic identities or theorems applicable to simplify the expression.
• Execute proper simplification steps and verify by expanding to check that the result matches the initial expression.

Factoring polynomial expressions can seem challenging at first, but with practice and awareness of identities and computational strategies, it becomes manageable. Understanding these concepts will solidify fundamental algebra skills.