Problem 55 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ 9 p^{2}-w^{8} $$

Step-by-Step Solution

Answer

(3p - w^4)(3p + w^4)

Step 1 - Recognize the Expression Type

Identify the expression as a difference of squares. The given polynomial is \(9p^2 - w^8\).

Step 2 - Write Each Term as a Square

Express each term as a square. \(9p^2 = (3p)^2\) and \(w^8 = (w^4)^2\).

Step 3 - Apply the Difference of Squares Formula

Use the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Thus, the expression can be written as: \((3p - w^4)(3p + w^4)\).

Step 4 - Check for Further Factorization

Verify if \((3p - w^4)\) or \((3p + w^4)\) can be factored further. In this case, both terms are prime polynomials.

Step 5 - Confirm the Final Factorization

Since \((3p - w^4)\) and \((3p + w^4)\) are prime, the final factorization is: \((3p - w^4)(3p + w^4)\).

Key Concepts

prime polynomials

When we talk about prime polynomials, we mean polynomials that cannot be factored further. In other words, a prime polynomial has no other factors aside from 1 and the polynomial itself.

For example, once we factorize the polynomial \(9p^2 - w^8\) into \((3p - w^4)(3p + w^4)\), each of these binomials is considered prime.

Always remember, just like prime numbers, prime polynomials are the simplest building blocks in polynomial factorization.

• Example: \(2x + 3\) is a prime polynomial because it cannot be simplified further.


• To check if a polynomial is prime, try factoring it. If you can't find any factors, then it is prime.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into a product of smaller polynomials.

This helps to simplify complex equations and can reveal important properties of the polynomial. Factorization can involve different techniques, such as using the greatest common factor, grouping terms, and recognizing special patterns.

• Different Methods: Identifying patterns like the difference of squares is key.
• Other methods include taking out common factors or re-grouping terms.

In our example, we used the difference of squares formula to factor \(9p^2 - w^8\) into \((3p - w^4)(3p + w^4)\).
Always ensure to check if the resulting factors can themselves be factored further.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. They form the basis of algebra and understanding them is crucial for mastering the subject.

They can range from simple to complex and knowing how to manipulate them is key.
Polynomial expressions are a significant subset of algebraic expressions.

• **Simple Example:** \(2x + 3\) where \(2x\) is a term and \(3\) is another.
• **Complex Example:** \(4x^2 + 7x + 5\) which includes a quadratic term, a linear term and a constant.

Working with algebraic expressions requires a solid understanding of basic operations, factorization techniques, and the properties of exponents.
Mastery of these components makes it easier to tackle more advanced problems in algebra.