Problem 36 - 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. $$12 x^{3}-27 x y^{2}$$

Step-by-Step Solution

Answer

The polynomial completely factors to \(3x(2x - 3y)(2x + 3y)\).

Step 1: Identify the Common Monomial Factor

Look for the greatest common factor (GCF) in the polynomial. For the terms \(12x^3\) and \(-27xy^2\), we notice that both can be divided by \(3x\). Thus, \(3x\) is the GCF.

Step 2: Factor Out the GCF

Divide each term in the polynomial by the GCF, \(3x\). This gives us:\[ \frac{12x^3}{3x} = 4x^2 \]\[ \frac{-27xy^2}{3x} = -9y^2 \]Thus, the polynomial \(12x^3 - 27xy^2\) can be rewritten as:\[ 3x(4x^2 - 9y^2) \]

Step 3: Recognize the Difference of Squares

Notice that \(4x^2 - 9y^2\) is a difference of squares, as \(4x^2 = (2x)^2\) and \(9y^2 = (3y)^2\). A difference of squares can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).

Step 4: Factor the Difference of Squares

Apply the difference of squares formula to \(4x^2 - 9y^2\):\[4x^2 - 9y^2 = (2x - 3y)(2x + 3y)\]Substitute back into the factored form with the GCF included:\[ 3x(2x - 3y)(2x + 3y) \]

Step 5: Final Check

Review each factor to ensure none can be further factored using integers. The expression \((2x - 3y)(2x + 3y)\) is fully factored, and \(3x\) is as simplified as possible. Therefore, the expression is completely factored.

Key Concepts

Common Monomial Factor

The process of factorization often begins with identifying a common monomial factor (CMF) across the terms of a polynomial. This step is crucial because factoring out the CMF simplifies the polynomial, making it easier to work with for further factorization. A common monomial factor essentially acts as the greatest divisor shared among all terms.
For instance, in the polynomial \(12x^3 - 27xy^2\), we first recognize the common monomial factor by examining each term individually. Here, both terms share \(3x\) as a factor. This means that each term can be divided by \(3x\) without leaving a remainder.

• In \(12x^3\), \(3x\) divides evenly, leaving behind \(4x^2\).
• In \(-27xy^2\), \(3x\) also divides evenly, resulting in \(-9y^2\).

Factoring this out, our polynomial simplifies to \(3x(4x^2 - 9y^2)\). Recognizing and extracting the CMF is vital as it reduces the size of the polynomial and paves the way for more advanced factoring techniques.

Difference of Squares

After factoring out the common monomial factor, the next step is to examine the remaining expression for any special patterns that may exist, such as the difference of squares. A difference of squares is a powerful algebraic identity that can simplify expressions significantly.
In mathematics, the difference of squares refers to an expression of the form \(a^2 - b^2\), which can be factored into \((a - b)(a + b)\).
Consider the expression obtained after factoring the CMF: \(4x^2 - 9y^2\). This aligns perfectly with the difference of squares pattern:

• \(4x^2\) can be rewritten as \((2x)^2\).
• \(9y^2\) can be expressed as \((3y)^2\).

Thus, \(4x^2 - 9y^2\) becomes \((2x - 3y)(2x + 3y)\). Using the difference of squares identity not only makes the polynomial simpler to handle but also makes it perfectly factored.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the highest number or expression that can divide all terms in a polynomial evenly. When factoring polynomials, determining the GCF is often the initial step before applying more intricate techniques.
To find the GCF of a polynomial, examine each term closely, breaking them down into their constituent factors.
In the polynomial \(12x^3 - 27xy^2\), breaking down each term reveals:

• \(12x^3\) consists of \(2\times 2 \times 3 \times x \times x \times x\).
• \(-27xy^2\) breaks down into \(-1 \times 3 \times 3 \times 3 \times x \times y \times y\).

From these factorizations, \(3x\) is identified as the greatest factor common to both terms. Dividing each term by this GCF simplifies the polynomial, laying a foundation for further factorization methods.
The GCF is a cornerstone in polynomial factorization, and finding it correctly can transform a complex expression into a much simpler one, as seen in our example where \(12x^3 - 27xy^2\) simplifies to \(3x(4x^2 - 9y^2)\).