Problem 58 - Elementary Algebra

Question

Factor completely. $$ 16 x_{4}-64 $$

Step-by-Step Solution

Answer

The completely factored form is \( 16(x^2+2)(x^2-2) \).

Step 1: Identify the Greatest Common Factor (GCF)

The expression given is \( 16x^4 - 64 \). First, we need to find the greatest common factor of both terms. In this case, both terms are divisible by 16. So, the GCF is 16.

Step 2: Factor Out the GCF

Factor out the GCF from the expression \( 16x^4 - 64 \). This gives us: \( 16(x^4 - 4) \).

Step 3: Recognize the Difference of Squares

Look at the expression inside the parentheses, \( x^4 - 4 \). This is a difference of squares because \( x^4 \) is \((x^2)^2\) and \( 4 \) is \(2^2\). Recall that the difference of squares formula is \( a^2 - b^2 = (a+b)(a-b) \).

Step 4: Apply the Difference of Squares Formula

Apply the difference of squares formula to factor \( x^4 - 4 \): \( x^4 - 4 = (x^2)^2 - (2^2) = (x^2 + 2)(x^2 - 2) \).

Step 5: Finalize the Factorization

Write down the complete factorization: \( 16(x^2+2)(x^2-2) \). This is the expression fully factored.

Key Concepts

Greatest Common Factor

When working with polynomials, finding the Greatest Common Factor (GCF) is a crucial first step in the factorization process. The GCF is the largest number or expression that divides all the terms within the polynomial without leaving a remainder.
In the problem given, we start with the expression \( 16x^4 - 64 \). The task is to determine what common factor exists between the terms. Both terms share a factor of 16.
Identifying the GCF means looking at both the numerical coefficient and the variable part. Since each coefficient can be divided by 16, and the variable \( x^4 \) doesn't affect the constant term -64, our GCF is clearly 16.

• Factor the polynomial by dividing each term by 16.
• Write the factored form as \( 16(x^4 - 4) \).

This step simplifies the expression and lays the groundwork for further factorization.

Difference of Squares

The next step in the factorization process often involves recognizing special patterns in polynomials. One such pattern is the Difference of Squares, which follows the formula \( a^2 - b^2 = (a+b)(a-b) \).
In our expression \( x^4 - 4 \), we identify it as a difference of squares. Notice that \( x^4 = (x^2)^2 \) and \( 4 = 2^2 \). Since our expression fits the pattern of \( a^2 - b^2 \), it can be rewritten using the difference of squares identity:

• Let \( a \) be \( x^2 \) and \( b \) be 2.
• Using the identity, \( (x^2)^2 - (2)^2 = (x^2 + 2)(x^2 - 2) \).

Recognizing and applying this formula is essential for simplifying polynomials efficiently.

Complete Factorization

Upon successfully factoring using the GCF and difference of squares, the equation is almost completely simplified. Complete factorization refers to breaking the expression down to its simplest form, where no further factorization is possible.
Our initial steps bring us to \( 16(x^2+2)(x^2-2) \), which is the fully factored version of the original polynomial. Let's break down what complete factorization entails:

• Ensure all factors are prime relative to each other, such as \( x^2 - 2 \) and \( x^2 + 2 \); no further factorization applies here.
• Check for any other special patterns or common factors, which aren't present in this case.

Now, the expression \( 16(x^2 + 2)(x^2 - 2) \) is the simplest form, with each component separated by multiplication, completing the factorization process.