Problem 16 - Elementary Algebra

Question

Factor completely. $$ 3 x 5-81 x 2 $$

Step-by-Step Solution

Answer

The completely factored form is \( 3x^2(x - 3)(x^2 + 3x + 9) \).

Step 1: Identify Common Factors

The given expression is \( 3x^5 - 81x^2 \). First, look for any common factors in both terms. Both terms have a factor of \( 3x^2 \).

Step 2: Factor Out the Greatest Common Factor

Factor \( 3x^2 \) out of the expression to get \( 3x^2(x^3 - 27) \). This is your expression after factoring out \( 3x^2 \).

Step 3: Recognize the Difference of Cubes

The expression inside the parenthesis \( x^3 - 27 \) is a difference of cubes since \( 27 \) can be written as \( 3^3 \). Thus, we have \( x^3 - 3^3 \).

Step 4: Apply the Difference of Cubes Formula

The difference of cubes formula is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). Use \( a = x \) and \( b = 3 \) to factor \( x^3 - 27 \) as \( (x - 3)(x^2 + 3x + 9) \).

Step 5: Write the Complete Factored Form

Combine the factored form from Step 2 and Step 4 to get the completely factored expression: \( 3x^2(x - 3)(x^2 + 3x + 9) \).

Key Concepts

Common Factors

When you're looking to factor an expression, the first thing you should do is find any common factors shared among its terms. This is known as the "greatest common factor" (GCF). In our original expression, which is \(3x^5 - 81x^2\), each term consists of products involving coefficients and variable powers. Both terms share a factor of \(3x^2\). By recognizing this, you can simplify the given expression by factoring \(3x^2\) out. Doing this reduces the complexity and helps you proceed to further factorization steps with ease. Also note that factoring out the GCF is always the initial step in any factorization problem as it simplifies the remaining expression right away.

Difference of Cubes

The expression inside the parentheses after factoring out the common factor becomes \(x^3 - 27\). This is a special type of polynomial known as the "difference of cubes." The key here is to understand that a cube refers to a number raised to the power of three. In our example, \(x^3\) is a cube (of \(x\)) and \(27\) is also a cube (of \(3\) since \(3^3 = 27\)).

This leads us to the "difference of cubes" identity: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a\) is \(x\) and \(b\) is \(3\). Using this identity, we can further break down the expression into \((x - 3)(x^2 + 3x + 9)\). This step helps make the factors visible and ready for any subsequent mathematical operations.

Factoring Polynomials

Factoring polynomials involves rewriting a given polynomial as a product of simpler factors, ideally polynomials of lower degrees. The process typically starts by identifying and factoring out common factors, as seen in the first step.

Further, you apply various working strategies like the difference of cubes, quadratics, or other identities depending on the structure of the polynomial's terms. As in our example, recognizing the difference of cubes was a crucial tactic.

• First, identify and factor out the GCF.
• Analyze the remaining polynomial to see if it's a special form (e.g., difference of cubes).
• Break it down using the appropriate formula.

Combining all these steps allows you to express the original polynomial as a product of its factors. In our problem, the completely factored form is \(3x^2(x - 3)(x^2 + 3x + 9)\). This demonstrates the ultimate goal of factoring - reducing a complex polynomial into simpler, more manageable pieces.