Problem 95 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 16 x^{3}-54 y^{3} $$

Step-by-Step Solution

Answer

The polynomial factors to \((2x - 3y)(4x^2 + 6xy + 9y^2)\).

Step 1 - Identify the common factor

Check if there is a greatest common divisor (GCD) for the coefficients and the variable terms in the polynomial. Here, the coefficients 16 and 54 have a GCD of 2, and there are no common variable terms.

Step 2 - Recognize the form of a difference of cubes

Rewrite the polynomial in the form of a difference of cubes. Notice that both 16 and 54 can be expressed as cubes: \[16 x^3 = (2x)^3 \] \[54 y^3 = (3y)^3 \]Therefore, the given expression can be rewritten as: \[(2x)^3 - (3y)^3 \]

Step 3 - Apply the difference of cubes formula

The formula for the difference of cubes is \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]Here, \(a = 2x\) and \(b = 3y\). Apply the formula:\[(2x)^3 - (3y)^3 = (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2) \]

Step 4 - Simplify the expression

Now, simplify the terms inside the parenthesis:\[(2x - 3y)(4x^2 + 6xy + 9y^2)\]Thus, the polynomial \(16 x^{3}-54 y^{3}\) factors to \( (2x - 3y)(4x^2 + 6xy + 9y^2) \).

Key Concepts

Greatest Common Divisor

When you start factoring a polynomial, the first step is often to identify the Greatest Common Divisor (GCD) of the coefficients. This is a number that divides each of the coefficients in the polynomial without leaving a remainder. For example, in the polynomial 16x^3 - 54y^3, the coefficients are 16 and 54.
The GCD of 16 and 54 is 2 because it is the largest number that can divide both of them evenly.
By factoring out the GCD, you make the subsequent steps easier. In this case, you factor out the 2 and proceed with the simplified polynomial.

Difference of Cubes

The difference of cubes is a special polynomial form. It allows you to factor expressions like a^3 - b^3. In our example, after factoring out the GCD, we get (2x)^3 - (3y)^3, which fits the difference of cubes pattern. The formula for factoring the difference of cubes is given by: a^3 - b^3 = (a - b)(a^2 + ab + b^2)This formula is incredibly useful because it breaks down the polynomials into simpler factors. For our polynomial, where a = 2x and b = 3y, we apply this formula to simplify further.

Factoring Techniques

Factoring polynomials involves breaking down a complex expression into simpler terms that can be multiplied to give the original expression. There are various techniques for this:

• Factoring by grouping.
• Factoring quadratics.
• Factoring special products like perfect squares and cubes

In the example, after applying the difference of cubes formula, we proceed to simplify the expression: (2x - 3y)(4x^2 + 6xy + 9y^2).This approach makes solving the polynomial much more straightforward and allows for easier identification of prime polynomials. A polynomial is considered prime if it cannot be factored any further. In this exercise, the factors obtained are as simple as they can get.