Problem 69 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ 25 u^{4}-81 z^{6} $$

Step-by-Step Solution

Answer

The factored form is \text{(5u^2 - 9z^3)(5u^2 + 9z^3)} and both factors are prime.

Step 1: Recognize as a difference of squares

Notice that the given polynomial is in the form of a difference of squares: 25u^4 - 81z^6

Step 2: Write each term as a square

Express each term as a square: 25u^4 = (5u^2)^2 81z^6 = (9z^3)^2

Step 3: Apply the difference of squares formula

Use the difference of squares formula, a^2 - b^2 = (a - b)(a + b) so: (5u^2)^2 - (9z^3)^2 = (5u^2 - 9z^3)(5u^2 + 9z^3)

Step 4: Determine if the factors are prime

Check if the factors can be factored further. (5u^2 - 9z^3) and (5u^2 + 9z^3) are already in their simplest forms and cannot be factored further.

Key Concepts

Difference of Squares

The 'difference of squares' is a special factoring technique. It deals with expressions that can be written as the difference (subtraction) of two perfect squares. A perfect square is a number or expression multiplied by itself. For example, 25 is a perfect square because it is equal to 5^2.

Here's the general form: a^2 - b^2 = (a - b)(a + b).

This equation says that any expression that is the difference of two squares can be factored into the product of two binomials: one with a sum and one with a difference. In our exercise, 25u^4 is (5u^2)^2 and 81z^6 is (9z^3)^2, making the polynomial 25u^4 - 81z^6 fit perfectly into the difference of squares formula.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored further over the set of integers or real numbers. Think of prime polynomials like prime numbers; they are 'building blocks' and can't be factored into simpler components.

For example, in the exercise provided, after applying the difference of squares formula, we get (5u^2 - 9z^3)(5u^2 + 9z^3). Both 5u^2 - 9z^3 and 5u^2 + 9z^3 are prime polynomials because they cannot be factored any further.

It's important to check for primality at the end of factoring polynomial problems to ensure that the factorization is in its simplest form.

Factoring Methods

Various methods are used to factor polynomials. Here are some common factoring methods:

• Greatest Common Factor (GCF): Find the largest term that divides all terms in the polynomial.
• Difference of Squares: As discussed, this method factors expressions in the form a^2 - b^2.
• Trinomials: Factoring trinomials involves finding two binomials that multiply to give the original trinomial.
• Grouping: This method groups terms to find common factors, then factors again if possible.

In the given problem, the difference of squares method was the most suitable. Recognizing when to use each method simplifies the process and leads to the correct solution faster. Remember, practice helps in identifying the best method to use.