Problem 66 - Elementary Algebra

Question

Factor completely. $$ 3 x_{4}-48 $$

Step-by-Step Solution

Answer

The expression is factored as \(3(x - 2)(x + 2)(x^2 + 4)\).

Step 1: Identify the Common Factor

The given expression is \(3x_4 - 48\). The first step in factoring is to identify the greatest common factor (GCF) of the terms in the expression. Notice that both terms, \(3x_4\) and \(-48\), have a common factor of 3. So, the GCF is 3.

Step 2: Factor Out the Greatest Common Factor

Once the GCF, which is 3, is identified, we factor it out from each term. Divide each term by 3:\[ 3x_4 - 48 = 3(x_4) - 3(16) = 3(x_4 - 16) \]Thus, the expression is factored as \(3(x_4 - 16)\).

Step 3: Identify If Further Factoring is Needed

The expression inside the parentheses, \(x_4 - 16\), should be checked for further factoring. \(x_4 - 16\) is a difference of squares since \(x_4\) is \((x^2)^2\) and 16 is \(4^2\). Recall the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).

Step 4: Apply the Difference of Squares Formula

Apply the difference of squares formula to \(x_4 - 16\). Set \(a = x^2\) and \(b = 4\), so:\[ x_4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) \]Since \(x^2 - 4\) can be factored further into \((x - 2)(x + 2)\), substitute it back:\[ x_4 - 16 = (x - 2)(x + 2)(x^2 + 4) \]

Step 5: Write the Complete Factored Expression

Based on the above factoring steps, substitute back into the original expression:\[ 3(x_4 - 16) = 3(x - 2)(x + 2)(x^2 + 4) \]This is the completely factored form of the original expression.

Key Concepts

Greatest Common Factor

When factoring polynomials, the first step is often to find the greatest common factor (GCF) of the terms involved. The GCF is the largest number or term that can evenly divide all terms in the expression. Identifying the GCF simplifies the polynomial, making further factoring more manageable.

• To find the GCF, look at the coefficients and variables of each term. For coefficients, it's simply the largest number that can divide each without a remainder.
• For variables, the GCF is each variable raised to the smallest power present in the expression.

For instance, in the expression \(3x_4 - 48\), we have the terms \(3x_4\) and \(-48\). The GCF of these is 3, because 3 is the largest number that divides both 3 and 48 evenly, and there are no common variables to consider. Once identified, you can factor out the GCF to simplify the expression.

Difference of Squares

The difference of squares is a specific pattern that appears in polynomials and is characterized by two terms, each of which is a perfect square, separated by a subtraction sign. The general formula is:

• \(a^2 - b^2 = (a - b)(a + b)\)

This formula states that a difference of two squares can be factored into a product of two binomials.
In the exercise, after factoring out the GCF, we are left with \(x_4 - 16\). Recognizing that \(x_4\) is \((x^2)^2\) and 16 is \(4^2\), we see this expression as a difference of squares:

• \(x_4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4)\)

Identifying and applying this pattern allows us to further factor the expression. The key is always to look for perfect squares within the expression.

Factoring Expressions

Factoring expressions involves breaking down a complex expression into simpler, multiplied factors. This process often consists of several steps, including finding the GCF and applying specific factoring formulas like the difference of squares.
After identifying and factoring out the GCF in the original expression \(3x_4 - 48\), we're left with \(x_4 - 16\). Applying the difference of squares formula results in \((x^2 - 4)(x^2 + 4)\). It's important to always check if further factoring is possible:

• \(x^2 - 4\) can be further factored using the difference of squares again: \((x - 2)(x + 2)\).

Combining all steps, the full factored expression is \(3(x - 2)(x + 2)(x^2 + 4)\). Appreciating the various factoring techniques will make the process systematic and clearer.