Problem 43 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$3 x^{4}-48$$

Step-by-Step Solution

Answer

The completely factored form is \(3(x - 2)(x + 2)(x^2 + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

In the polynomial \(3x^4 - 48\), we first look for a common factor in both terms. Both terms are divisible by 3, which is the greatest common factor (GCF). Factoring out the GCF gives us:\[ 3(x^4) - 3(16) \] which simplifies to \[ 3(x^4 - 16) \].

Step 2: Recognize the Difference of Squares

Observe that the expression inside the parentheses \(x^4 - 16\) is a difference of squares. It can be rewritten as \((x^2)^2 - 4^2\). The difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\) allows us to factor it further.

Step 3: Apply the Difference of Squares Formula

Apply the difference of squares formula to \(x^4 - 16\). Now, factor it as:\[ (x^2 - 4)(x^2 + 4) \]. This gives us the factorization: \[ 3(x^2 - 4)(x^2 + 4) \].

Step 4: Factor Again (if possible)

Check the new factors from Step 3. The expression \(x^2 - 4\) is another difference of squares, as it can be written as \((x)^2 - (2)^2\). This can be factored further to \((x - 2)(x + 2)\). Thus, the polynomial becomes:\[ 3(x - 2)(x + 2)(x^2 + 4) \]. The term \(x^2 + 4\) is not factorable over the integers as it is a sum of squares.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is crucial for simplifying polynomials, especially when tackling factorization tasks. The GCF refers to the largest number, including polynomials, that can evenly divide each term in the expression. Let's break this down a bit more!

• **Finding the GCF:** Look at the coefficients and variables of the polynomial terms. For instance, in the polynomial \(3x^4 - 48\), both terms share a factor of 3. This makes 3 the GCF.
• **Factoring the Polynomial:** By factoring the GCF from each term, you simplify the polynomial significantly. In the example, factoring out 3 transforms the original expression into \(3(x^4 - 16)\), which is more manageable.

Identifying and factoring out the GCF is often the first step in polynomial factorization, allowing you to tackle more complex forms more efficiently further along.

Difference of Squares

The difference of squares is a powerful tool in algebra, frequently used to simplify terms by expressing polynomials as a product of two binomials. This form is present when a polynomial fits the pattern \(a^2 - b^2\).

• **Identifying the Difference:** Observe terms like \(x^4 - 16\) which can be rewritten as \((x^2)^2 - 4^2\), perfectly matching the difference of squares format.
• **Using the Formula:** The fundamental formula \(a^2 - b^2 = (a-b)(a+b)\) allows you to express \((x^2)^2 - 4^2\) as \((x^2 - 4)(x^2 + 4)\), simplifying further.

Understanding and employing the difference of squares helps break down complex polynomials into smaller, more solvable parts, making it an indispensable tool for students.

Sum of Squares

The sum of squares frequently confuses students, as it cannot be factored over the integers like the difference of squares. In mathematics, expressions like \(a^2 + b^2\) require a different approach since they resist factorization into real numbers.

• **Recognizing the Limitation:** Unlike \(a^2 - b^2\), \(x^2 + 4\) cannot be easily factored into real integer terms.
• **Complex Numbers Approach:** While not necessary for this context, it's worth noting that in some advanced classes, factors involve complex numbers: \(a^2 + b^2\) can be seen as \((x + 2i)(x - 2i)\).

Polynomial factorization often ends when a sum of squares is encountered, as no integer-based factorization is typically possible, highlighting a unique element in algebra that is important to recognize.