Problem 66 - College Algebra

Question

Factor completely, or state that the polynomial is prime. $$ 5 x^{3}-45 x $$

Step-by-Step Solution

Answer

The complete factorization of the given polynomial \(5 x^{3}-45 x\) is \(5x(x-3)(x+3)\).

Step 1: Identify common factors

First, identify if there is a common factor in all terms of the polynomial. In this given polynomial \(5 x^{3}-45 x\), we notice that each term is divisible by \(5x\).

Step 2: Factor out the common factor

The next step is to factor out this common factor. If we do that, we get \(5x(x^{2}-9)\).

Step 3: Identify additional factoring

Now, let's take a closer look at \(x^{2}-9\). This is a difference of squares, which is a special type of binomial that can be factored further.

Step 4: Factorize the difference of squares

We can write \(x^{2}-9\) as \((x-3)(x+3)\). Marshaling all these factors together, we get \(5x(x-3)(x+3)\).

Key Concepts

Difference of Squares

When we talk about the difference of squares in algebra, we refer to expressions of the form \(a^2 - b^2\). This special form is notable because it can be factored into the product of two distinct linear factors, \(a - b\) and \(a + b\).

For instance, in the provided exercise, the expression \(x^2 - 9\) is identified as a difference of squares. Why? Because it can be represented as \(x^2 - 3^2\), which clearly matches the \(a^2 - b^2\) template where \(a = x\) and \(b = 3\). After recognizing this pattern, we can apply the factoring formula to yield \(x - 3\) and \(x + 3\). By rewriting and simplifying this expression, the solution becomes much clearer and the problem is effectively simplified.

Common Factors in Polynomials

Identifying common factors plays an important role in simplifying algebraic expressions and factoring polynomials. A common factor is a term that divides evenly into each term of the polynomial. In this case, the polynomial \(5 x^3 - 45 x\) has a common factor of \(5x\) that can be factored out.

By extracting the common factor from each term, we're left with a simpler expression. This initial step is vital as it can unveil further factorization opportunities, like revealing a difference of squares or setting the stage for factoring by grouping. The process not only simplifies calculations but can also shed light onto the structure of the polynomial itself.

Prime Polynomial

In algebra, a prime polynomial can be likened to a prime number in arithmetic—it cannot be factored into the product of two non-constant polynomials. Determining whether a polynomial is prime is a matter of attempting to factor it.

If no common factors are apparent and no factorization techniques apply (such as factoring a difference of squares or applying the quadratic formula), the polynomial is then considered prime. In our exercise, before the factoring steps, you might wonder if the polynomial is prime. However, upon closer inspection, we find that there are indeed common factors and the structure of a difference of squares, demonstrating that the polynomial is not prime.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). They're the cornerstone of algebra and are what we manipulate through various operations and factoring techniques.

In our polynomial factoring exercise, we've started with a complex algebraic expression \(5x^3 - 45x\) and simplified it by recognizing patterns, extracting common factors, and finally using the difference of squares rule. Understanding and manipulating these expressions is key to solving algebraic problems and essential in progressing through more advanced mathematical concepts.