Problem 53 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ a u^{2}+(a+c) u+c $$

Step-by-Step Solution

Answer

The short answer to the question is: The factored form of the polynomial \(a u^{2}+(a+c) u+c\) is \(u(au +(a+c) + \frac{c}{u})\).

Step 1: Identify common factors

Given the polynomial \(a u^{2}+(a+c) u+c\), we first need to identify any common factors among the terms. In this case, each term contains the variable 'u', which can be factored out.

Step 2: Factor out the common factor 'u'

Factoring out the common factor 'u', we get: $$ u(au +(a+c) + \frac{c}{u}) $$

Step 3: Check if it can be factored further

Now, let's examine the polynomial inside the parenthesis: \(au +(a+c) + \frac{c}{u}\). We notice that there are no more common factors among the terms, and no other recognizable factoring techniques can be applied to simplify it further. Therefore, the polynomial is now factored and cannot be simplified any further.

Step 4: State the final factored polynomial

The factored form of the given polynomial is as follows: $$ u(au +(a+c) + \frac{c}{u}) $$

Key Concepts

Common Factor

Polynomial factoring begins with identifying the common factor in a given expression. A common factor is a term that is present in all parts of the polynomial. By recognizing this shared factor, you simplify your task of factoring.
In the original polynomial, \( a u^{2} + (a+c) u + c \), we identify 'u' as a common factor because it appears at least once in every term. To simplify the polynomial, you extract (or factor out) this 'u' from each term. This reduces the polynomial to a simpler form.
Finding common factors is like identifying a pattern. It's a crucial first step in many algebraic operations. Remember, the common factor can be a number, variable, or both.

Factoring Techniques

Once common factors are handled, you move on to other factoring techniques. Factoring is the process of breaking down a complex expression into simpler factors that, when multiplied together, give back the original expression. Polynomials can sometimes require a variety of techniques to be fully factored.
Here are some techniques:

• Grouping: Sometimes terms in a polynomial can be classified into groups, which can then be factored separately to simplify the polynomial.
• Quadratic factoring: This is used for expressions in the form \(ax^2 + bx + c\). Often involves finding the roots or using the quadratic formula.
• Special products: Recognizing patterns such as perfect square trinomials or difference of squares can help in factoring.

In the case of our polynomial \( u(au +(a+c) + \frac{c}{u}) \), we attempt other techniques only when common factors are exhausted. If none apply, we consider other properties like being a prime polynomial.

Prime Polynomial

A prime polynomial is analogous to a prime number. It's a polynomial that cannot be factored further over the set of integers. In simpler terms, it's a polynomial that you can't break down into simpler polynomials with integer coefficients.
When factoring does not yield any additional insight or simpler expression, the polynomial may itself be a prime polynomial. This was the case with the expression \(u(au +(a+c) + \frac{c}{u})\) once we factored out the common factor 'u'.
Understanding when a polynomial is truly prime requires familiarity with various factoring techniques. If all methods have been exhausted and no simpler form is found, it is acknowledged as prime. Recognizing prime polynomials helps in solving problems efficiently by knowing when to stop factoring and move forward with other steps in algebra.