Problem 34 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$3 x^{3}+15 x$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial is \(3x(x^2+5)\).

Step 1: Identify Common Factors

First, observe the terms in the polynomial. It can be noticed that both terms share a common factor of 3x.

Step 2: Factor out Common Factors

Once the common factor is identified, the next step is to factor it out of each term in the polynomial. This gives us \(3x(x^2+5)\).

Step 3: Check for Further Factorization

Lastly, check if the remaining part of the polynomial \(x^2 + 5\) can be factored any further. Since there are no more common factors that can be factored out, the polynomial is completely factored.

Key Concepts

Common Factors in Polynomials

Understanding the concept of common factors in polynomials is essential for simplifying algebraic expressions effectively. It is similar to finding common ground between different numbers, but in this case, the numbers are parts of an algebraic expression. Initiate by examining each term to spot any shared factors. In the exercise given, both terms of the polynomial, which are \(3x^3\) and \(15x\), share a factor of \(3x\).

Finding the greatest common factor (GCF) here is pivotal. It's akin to asking what's the most substantial part you can remove from each term before they start looking unique. Once the common factor is spotted, you painstakingly pull it out from the polynomial like a thread from fabric, simplifying the expression. This process is comparable to decluttering a room; you remove what’s common—say, piles of books—so you can clearly see and manage the remaining items individually. In the case of our polynomial, extracting the common factor simplifies it to \(3x(x^2+5)\).

Polynomial Prime Factorization

Polynomial prime factorization is a bit of a treasure hunt. Imagine you're looking for hidden gems within an expression that at first glance, might look complex or intimidating. It involves breaking down the polynomial into the product of its prime factors, which are the individual, unbreakable pieces of the algebraic puzzle. In the context of our example, once we factored out the common factor, we need to inspect the remaining expression, \(x^2+5\), to see if it can be cracked open any further.

Regrettably, this is where the search ends in this case. After scouting for any other common factors, and finding none, you can confidently declare that what you're left with is as simplified as it gets—there are no smaller, ‘prime’ pieces to extract. It's like confirming that there are no more gold nuggets in the stream. Therefore, the conclusion is that the polynomial \(3x(x^2+5)\) is in its prime factorized form, with \(x^2+5\) being an irreducible polynomial—much like a prime number in the realm of integers.

Algebraic Expressions

Algebraic expressions are the backbone of algebra and are essentially phrases in the language of mathematics. They can be simple or complex, consisting of numbers, variables, and arithmetic operations. Think of them as sentences, where variables and coefficients are the words, while addition, subtraction, multiplication, and division are the grammar rules that bind them together.

In our exercise, the algebraic expression began as \(3x^3 + 15x\). This expression tells a story of two quantities being added together. The variables, which are the 'x’s, represent a certain unknown value, and they come along with their numerical buddies, the coefficients, which in this case are 3 and 15. To simplify these expressions, just like editing a verbose paragraph, you find redundancies or commonalities, factor them out, and end up with a more concise, understandable piece—here, becoming \(3x(x^2+5)\). Thus, it should be your goal to express these algebraic sentiments as eloquently and succinctly as possible.