Problem 43 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ 8 h^{3}-27 k^{3} $$

Step-by-Step Solution

Answer

(2h - 3k)(4h^2 + 6hk + 9k^2) is the factored form, with 4h^2 + 6hk + 9k^2 being a prime polynomial.

Step 1: Recognize the form

The expression given is a difference of cubes. Identify the standard formula for factoring a difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2) In this case, a^3 = 8h^3 and b^3 = 27k^3.

Step 2: Express each term as a cube

Express each term in the format of cubes: 8h^3 = (2h)^3 27k^3 = (3k)^3 This means a = 2h and b = 3k.

Step 3: Apply the difference of cubes formula

Substitute a and b into the difference of cubes formula: 8h^3 - 27k^3 = (2h - 3k)[(2h)^2 + (2h)(3k) + (3k)^2] This simplifies to: (2h - 3k)(4h^2 + 6hk + 9k^2)

Step 4: Check for prime polynomials

Examine the factored form (2h - 3k)(4h^2 + 6hk + 9k^2). The second factor, 4h^2 + 6hk + 9k^2, does not factor further and is therefore a prime polynomial.

Key Concepts

difference of cubes

When it comes to factoring polynomials, understanding special products like the difference of cubes can make the process simpler. A difference of cubes is any expression of the form \(a^3 - b^3\), and it can be factored using a specific formula.
The standard formula for factoring a difference of cubes is:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
This formula allows you to break down the polynomial into a product of two binomials, making it easier to handle.
For example, consider the expression \(8h^3 - 27k^3\). First, we can express each term as a cube: \(8h^3 = (2h)^3\) and \(27k^3 = (3k)^3\). Here, \(a = 2h\) and \(b = 3k\).
Applying the difference of cubes formula:
\[8h^3 - 27k^3 = (2h - 3k)((2h)^2 + (2h)(3k) + (3k)^2)\]
This simplifies to:
\[(2h - 3k)(4h^2 + 6hk + 9k^2)\]
By recognizing and applying the difference of cubes formula, the original polynomial is effectively factored.

prime polynomials

A prime polynomial is one that cannot be factored further over the set of integers. These are similar to prime numbers in arithmetic. To identify a prime polynomial, you must ensure it doesn't have any factors other than itself and 1.
Take the factored expression from our earlier example: \[(2h - 3k)(4h^2 + 6hk + 9k^2)\].
Here, the first factor, \(2h - 3k\), is a simple binomial. The second factor, \(4h^2 + 6hk + 9k^2\), is a trinomial.
The trinomial \((4h^2 + 6hk + 9k^2)\) represents a prime polynomial because it cannot be factored further into simpler polynomials. We know it is prime because no two binomials multiplied together can result in this trinomial without involving fractions or complex numbers.
Identifying prime polynomials helps in fully simplifying and understanding polynomial expressions.

factoring formulas

Factoring formulas are essential tools in algebra that allow us to break down complex polynomials into simpler components. One of the most commonly used formulas is the difference of cubes:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
Such formulas help in transforming a challenging expression into a product of polynomials that are easier to manage.
Aside from the difference of cubes, other notable factoring formulas include:

• The sum of cubes: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
• Difference of squares: \[a^2 - b^2 = (a - b)(a + b)\]
• Perfect square trinomials: \[a^2 + 2ab + b^2 = (a + b)^2\] and \[a^2 - 2ab + b^2 = (a - b)^2\]

Knowing and regularly practicing these formulas can significantly simplify the process of factoring.
Whenever you encounter a polynomial, try to match it with one of these formulas. With time, recognizing these patterns becomes second nature, enabling quicker and more accurate solutions.