Problem 32 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$x^{2}+4$$

Step-by-Step Solution

Answer

The polynomial \(x^{2}+4\) is prime. It cannot be factored using real numbers.

Step 1: Check if the Polynomial is in a Recognizable Form

To begin, we check if the given polynomial is presented in any recognizable form or if any known identities fit into it. Unfortunately, it doesn't look like \(x^{2}+4\) fits into any of standard identities. It isn't in the format of \(a^2 - b^2\), \(a^3 \pm b^3\) or \(a^2 \pm 2ab + b^2\).

Step 2: Check If the Polynomial Can Be Factored

There are no values of x that would result in zero when plugged into \(x^{2}+4\). Hence, the polynomial cannot be factored using real numbers.

Step 3: Confirm if the Polynomial is Prime

As it has been established now that this polynomial can't be factored further using real numbers, it's established that it is a prime polynomial.

Key Concepts

Polynomial Prime

When we say a polynomial is prime, we mean that it cannot be factored into the product of two non-constant polynomial factors with real coefficients. This is similar to prime numbers, which are numbers that have no divisors other than 1 and themselves. For instance, consider the polynomial given in the exercise, \(x^{2}+4\). After analyzing it with the purpose of factoring, we see that it does not exhibit roots that are real numbers and therefore it cannot be broken down into more basic polynomial factors over the set of real numbers.

In general, a polynomial could be prime for several reasons. It might not fit any of the recognizable polynomial forms suitable for factoring, or it might not have integer or rational roots. Recognizing a prime polynomial is essential as it informs us that attempting to factor it further within the real numbers is futile, and can save us time when solving algebra problems.

Recognizable Polynomial Forms

Certain polynomials follow specific forms that make them easier to factor. These recognizable polynomial forms often reflect well-known algebraic identities. For example, difference of squares \(a^2 - b^2 = (a + b)(a - b)\), perfect square trinomials \(a^2 \pm 2ab + b^2 = (a \pm b)^2\), and sum/difference of cubes \(a^3 \pm b^3\) are all patterns that can greatly simplify factoring when they are identified. In the given exercise, the polynomial \(x^{2}+4\) was examined for such forms. However, it does not match any of them, as it is a sum of squares rather than a difference, and cannot be represented as a sum or difference of cubes or a perfect square trinomial. This observation is essential in determining that the polynomial does not succumb to these typical factoring techniques and therefore must be approached differently or declared prime in context to real numbers.

Factoring Algebraic Expressions

Factoring an algebraic expression means writing it as the product of its factors. This process is vital in solving equations, simplifying expressions, and finding roots. There are several factoring techniques:

• Factoring out the greatest common factor (GCF).
• Factoring by grouping.
• Using special formulas, like the difference of squares.
• Finding the roots of the polynomial and factoring by its zeros.

Despite these techniques, not all polynomials are factorable in the realm of real numbers. The exercise \(x^{2}+4\) is an example of a polynomial without real roots, leading us to the conclusion that it cannot be factored using traditional real-number methods. In such cases, recognizing a polynomial is prime is key to solving or simplifying expressions containing it. Advanced mathematical topics, such as complex numbers, may allow for further factoring, but within the scope of real numbers, our exploration halts at this juncture.