Problem 99 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 9 x^{3}+36 y^{3} $$

Step-by-Step Solution

Answer

9(x + \sqrt[3]{4}y)(x^2 - x(\sqrt[3]{4}y) + (\sqrt[3]{4}y)^2).

Step 1: Identify Common Factors

Look for any common factors in the terms of the polynomial. Here, the terms are \(9x^3\) and \(36y^3\). Both terms have a common factor of 9.

Step 2: Factor Out the Common Factor

Factor out the common factor of 9 from both terms: \[9(x^3 + 4y^3)\]

Step 3: Recognize the Sum of Cubes

The expression inside the parentheses, \(x^3 + 4y^3\), is a sum of cubes. Sum of cubes can be factored using the formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) where \(a = x\) and \(b = (4y)^{1/3}\).

Step 4: Apply the Sum of Cubes Formula

Apply the sum of cubes formula to \(x^3 + (4y^3)\). Since the cube root of 4 is \(\sqrt[3]{4}\), we get: \[x^3 + 4y^3 = (x + \sqrt[3]{4}y)(x^2 - x(\sqrt[3]{4}y) + (\sqrt[3]{4}y)^2)\].

Step 5: Combine All Factors

Multiply the common factor back into the factored sum of cubes: \[9(x + \sqrt[3]{4}y)(x^2 - x(\sqrt[3]{4}y) + (\sqrt[3]{4}y)^2)\]

Step 6: Identify Prime Polynomial

Check if any polynomials in the final expression cannot be factored further. Here, no further factoring can be done, so the expression is fully factored.

Key Concepts

sum of cubes

The sum of cubes is an important concept in polynomial factoring. When you have two cubed terms added together, like in the expression \(a^3 + b^3\), you can factor it using a special formula. This formula is:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
For example, consider the expression from the exercise: \(x^3 + 4y^3\). To use the sum of cubes formula:

• Identify the cubed terms: \(x^3\) and \(4y^3\).
• Find the cube roots: \(x\) and \(\sqrt[3]{4}y\).
• Apply the formula: \(x^3 + (\sqrt[3]{4}y)^3\).

This gives us:
\((x + \sqrt[3]{4}y)(x^2 - x\sqrt[3]{4}y + (\sqrt[3]{4}y)^2)\).
This formula helps simplify complex polynomial expressions into a product of simpler polynomials, making it easier to analyze and solve.

common factors

Before you start factoring any polynomial, it's smart to look for common factors. These are numbers or expressions that evenly divide all terms in the polynomial.
In the original problem, we had: \(9x^3 + 36y^3\). To find the common factor:

• Identify the greatest common divisor (GCD) of the coefficients: 9 and 36. The GCD is 9.
• Check if any variables are common in all terms; in this case, none.

So, we factor out the common factor of 9:
\(9(x^3 + 4y^3)\).
Factoring out common factors simplifies the polynomial and makes subsequent factoring steps much easier.

prime polynomials

A prime polynomial is one that cannot be factored further using real numbers. Identifying these can save time because you’ll know the polynomial is fully simplified.
Consider the last steps for the expression: after factoring the common factor and applying the sum of cubes formula, we got:

• \(9(x + \sqrt[3]{4}y)(x^2 - x\sqrt[3]{4}y + (\sqrt[3]{4}y)^2)\).

At this stage, check if any polynomial part can be factored further. For this exercise, the factors: \(x + \sqrt[3]{4}y\) and \((x^2 - x\sqrt[3]{4}y + (\sqrt[3]{4}y)^2)\) are prime.
Therefore, the fully factored form is:
\(9(x + \sqrt[3]{4}y)(x^2 - x\sqrt[3]{4}y + (\sqrt[3]{4}y)^2)\).
Knowing when you've reached a prime polynomial ensures you don't spend unnecessary time trying to factor expressions further.