Problem 20 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$6 x^{3}+24 x$$

Step-by-Step Solution

Answer

The factorization of \(6x^3 + 24x\) is \(6x * (x^2 + 4)\).

Step 1: Identify common factors

Remember that whenever you want to factorize a polynomial, always start by looking for common factors. In this case, observe that both terms in the polynomial, \(6x^3\) and \(24x\), can be divided by \(6x\). This is your common factor.

Step 2: Factorize the Polynomial

Now that you've identified the common factor, you can use this to factorize your polynomial. The polynomial \(6x^3 + 24x\) can be factorized as \(6x * (x^2 + 4)\).

Step 3: Check the result

To ensure that the factorization is done correctly, multiply the factored terms again to see if we obtain the original polynomial. Multiply \(6x\) by each term in the brackets \(6x * (x^2 + 4) = 6x^3 + 24x\). Since this matches with our original polynomial, we can say that our factorization is correct.

Key Concepts

Common Factors in Polynomials

In algebra, identifying common factors in polynomials is a fundamental skill. It's like finding what ingredients are shared in different recipes. Let's take the polynomial in our exercise, which is 6x^3 + 24x. Think of these terms as different recipes that happen to use some of the same ingredients - in this case, numbers and variables that are present in all terms.

When you look at this polynomial, both terms include a multiple of 6 and the variable x. This means that 6x is a 'common ingredient' and is referred to as the common factor.

To effectively use common factors for simplifying expressions, you first identify them by examining what numbers and variables each term of the polynomial share. It's a bit like a game of 'spot the difference,' except you're spotting the similarities. This step is crucial because it simplifies the polynomials, making further factoring much easier and clearing the path towards finding the simplest form of the expression.

Polynomial Factorization Techniques

Polynomial factorization can be likened to dismantling a complex machine into its basic components. There are various techniques to factorize polynomials, each akin to using different tools for different kinds of machinery.

Distributive Property

Also known as the 'reverse of distribution,' this method involves taking out the common factor and rewriting what remains inside parentheses, just as we did in our exercise with 6x^3 + 24x. It’s like removing what’s common and seeing what’s left.

Factoring by Grouping

Sometimes, polynomials have four or more terms, and you have to group them to find common factors. It's like organizing a toolbox—group similar tools together to make them easy to find.

Special Products

Certain polynomials are the result of multiplying special binomials like the difference of squares or perfect square trinomials. Recognizing these patterns is like identifying a type of engine model by its components.

Quadratic Polynomials

Polynomials with a degree of 2 may be factorized by finding two numbers that multiply to give the constant term while adding to give the coefficient of the middle term. It's like solving a riddle with specific number clues.

Each technique is a way of unpacking the polynomial into simpler factors, leading us gradually to the most streamlined form of the expression.

Checking Polynomial Factorization

Just like a chef tastes a dish before serving to ensure quality, checking your polynomial factorization is essential to verify accuracy. After you’ve factorized a polynomial, how do you know it’s done correctly? One reliable method is to perform a multiplication check.

Let’s use the product from our exercise, 6x * (x^2 + 4), as an example. To check it, simply multiply the factored terms and see if you return to the original polynomial. If we distribute 6x into the parentheses, multiplying it with each term inside, we should get 6x^3 + 24x—the original problem.

Another method is to use graphing utility software to compare graphs before and after factoring. The graphs should coincide because factoring changes the expression's format, not its value.

Remember, checking your work confirms that you’ve factorized without altering the polynomial's true nature - it's an essential step to ensure your solution is both accurate and reliable.