Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ u^{4}-v^{4} $$

Step-by-Step Solution

Answer

The polynomial \(u^4 - v^4\) can be factored using the difference of squares formula. The final factored form is \((u - v)(u + v)(u^2 + v^2)\).

Step 1: We can see that the given polynomial is \(u^4 - v^4\), where \(u^4\) is a square and \(v^4\) is another square. #Step 2: Apply the difference of squares formula#

Recall the difference of squares formula: \((a^2 - b^2) = (a - b)(a + b)\). In our case, we have \(a = u^2\) and \(b = v^2\). Apply the formula to our polynomial: \[ (u^2 - v^2)(u^2 + v^2) \] #Step 3: Factor the first term further#

Step 2: We can see that \((u^2 - v^2)\) is also a difference of squares. We can apply the formula again with \(a = u\) and \(b = v\): \[ (u - v)(u + v)(u^2 + v^2) \] #Step 4: The final factored form#

Our polynomial has now been completely factored as: \[ (u - v)(u + v)(u^2 + v^2) \] This is the final factored form of the given polynomial \(u^4 - v^4\).

Key Concepts

Difference of Squares

The difference of squares is a key concept in polynomial factorization. It is a technique used to factor expressions of the form \(a^2 - b^2\). This type of polynomial can be factored into the product of two binomials, specifically \((a - b)(a + b)\). The essential idea is that the square terms can be decomposed into roots, which, when multiplied, reproduce the expression.

For instance, in the expression \(u^4 - v^4\), both terms \(u^4\) and \(v^4\) are perfect squares, with their square roots being \(u^2\) and \(v^2\) respectively. By applying the difference of squares formula, this polynomial can be initially factored as \((u^2 - v^2)(u^2 + v^2)\). The difference of squares formula is a fundamental bridging step in many polynomial factorization problems.

When further inspecting \(u^2 - v^2\), we notice it also represents a difference of squares. Applying the technique again reveals another layer of factors: \((u - v)(u + v)\). This showcases how powerful and recurring the difference of squares method is in simplifying and factoring complex polynomials.

Factoring Techniques

Factoring techniques involve various methods used to simplify polynomials by writing them as a product of simpler expressions or factors. The difference of squares is just one such technique.

Different polynomials require different approaches depending on their structure. Here are some common techniques:

• **Greatest Common Factor (GCF)**: Extracting the largest factor common to all terms.
• **Grouping**: Rearranging and combining terms to factor by grouping.
• **Trinomials and special forms**: Factoring using special patterns and formulas.
• **Difference of squares, sum of squares, and sum/difference of cubes**: Special cases that each have distinctive formulas.

In the given exercise, after identifying \(u^4 - v^4\) as a difference of squares, the specific technique involves applying the difference of squares formula twice. This results in expressions that are easier to manage and solve. Consistently using appropriate techniques can simplify polynomial expressions drastically, making it easier to solve related equations.

Prime Polynomial

A prime polynomial is one that cannot be factored into simpler polynomials with integer coefficients. Similar to prime numbers, which have no divisors other than 1 and themselves, a prime polynomial is in its simplest form and cannot be broken down into a product of polynomials.

The exercise demonstrates how to ascertain if a polynomial is prime. After completely factoring \(u^4 - v^4\), we realize that our expression ends up as \((u - v)(u + v)(u^2 + v^2)\). Here, \(u^2 + v^2\) cannot be factored further over the real numbers, implying it remains prime in this context.

Understanding when a polynomial is prime helps in quickly identifying factorizable components, thus streamlining problem-solving processes, especially in advanced algebra contexts. Recognizing a prime polynomial involves observing the polynomial’s structure and determining if further factorization is possible. This understanding is crucial for effectively managing polynomial expressions.