Problem 101 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 27 h^{3}-k^{3} $$

Step-by-Step Solution

Answer

Prime factors are \(3h - k\) and \(9h^2 + 3hk + k^2\).

Step 1 - Identify the Form

The expression is in the form of a difference of cubes: \(a^3 - b^3\), where \(a = 3h\) and \(b = k\).

Step 2 - Write the Difference of Cubes Formula

The formula for the difference of cubes is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Step 3 - Substitute the Values

Substitute \(a = 3h\) and \(b = k\) into the formula: \(27h^3 - k^3 = (3h - k)((3h)^2 + (3h)(k) + k^2)\).

Step 4 - Simplify

Simplify the terms inside the second parenthesis: \((3h)^2 = 9h^2\), \( (3h)(k) = 3hk\), and \( k^2\). So, the expression becomes: \(27h^3 - k^3 = (3h - k)(9h^2 + 3hk + k^2).\)

Step 5 - Check for Prime Polynomials

The factors \(3h - k\) and \(9h^2 + 3hk + k^2\) cannot be factored further. Therefore, they are prime polynomials.

Key Concepts

difference of cubes

The 'difference of cubes' is a concept in algebra that refers to the subtraction of one cube from another. In general form, it looks like this: \(a^3 - b^3\). This form is special because it can be factored using a straightforward formula:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
For instance, in the given exercise, the expression \(27h^3 - k^3\) is identified as a 'difference of cubes,' with \(a = 3h\) and \(b = k\). The formula allows us to break down this seemingly complex expression into simpler factors, making it easier to deal with in further calculations.

prime polynomials

Prime polynomials are polynomials that cannot be factored any further over the set of integers. They are similar to prime numbers in arithmetic, which cannot be divided evenly by any other number except one and themselves. After factoring an expression, the resulting factors should be checked to see if they can be factored down even more. If they cannot, they are considered prime polynomials.
In the example \(27h^3 - k^3\), after applying the difference of cubes formula, we obtain:
\[(3h - k)(9h^2 + 3hk + k^2)\]
Neither \(3h - k\) nor \(9h^2 + 3hk + k^2\) can be factored further, meaning these are prime polynomials.

polynomial factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. This process helps in solving equations and simplifying expressions, making it a fundamental tool in algebra. There are various methods to factor polynomials, including factoring by grouping, using special formulas like the difference of squares or cubes, and the general process of finding roots.
In our given problem, we start by identifying that the expression \(27h^3 - k^3\) is a difference of cubes. Using the corresponding formula, we factor the polynomial into simpler components:
\[(3h - k)(9h^2 + 3hk + k^2)\]
This decomposition allows for easier manipulation in further mathematical operations. Factorization not only simplifies solving equations but also makes understanding the underlying structure of polynomials more transparent and manageable.