Problem 51 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ x^{4}+3 x^{3}-2 x-6 $$

Step-by-Step Solution

Answer

The short answer for the given polynomial \(x^4 + 3x^3 - 2x - 6\) after factoring is: \((x+3)(x-\sqrt[3]{2})(x^2 + x\sqrt[3]{2} + (\sqrt[3]{2})^2)\)

Step 1: Identify Potential Patterns

We can notice that the polynomial has all powers of x from 0 to 4 (i.e., there is no missing power) and the coefficients are integers. So, we will proceed with factoring by grouping technique.

Step 2: Grouping

First, we will group the polynomial into two pairs: \((x^4 + 3x^3)\) and \((-2x - 6)\).

Step 3: Factor Each Group

Next, we will factor out the Greatest Common Factor (GCF) from each group: From the first group (\(x^4 + 3x^3\)), we can factor out \(x^3\), giving us: \(x^3(x+3)\). From the second group (\(-2x - 6\)), we can factor out \(-2\), giving us: \(-2(x+3)\). Now, rewrite the polynomial with these factors: \(x^3(x+3) - 2(x+3)\).

Step 4: Factor Out the Common Binomial

Notice that both terms have a common factor of \((x+3)\). We can now factor this out from the expression: \((x+3)(x^3 - 2)\).

Step 5: Check if Further Factoring is Possible

Now we have factored the polynomial as \((x+3)(x^3 - 2)\). We should check if further factoring is possible for \(x^3 - 2\). Since this expression is a difference of cubes, we can factor it further using the following formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) In our case, \(a = x\) and \(b= \sqrt[3]{2}\). So we can factor \(x^3-2\) as: \((x-\sqrt[3]{2})\)(\(x^2 + x\sqrt[3]{2} + (\sqrt[3]{2})^2\))

Step 6: Write the Final Factored Form

Now that we have factored all the terms possible, we can write the final factored form of the given polynomial: \((x+3)(x-\sqrt[3]{2})(x^2 + x\sqrt[3]{2} + (\sqrt[3]{2})^2)\)

Key Concepts

Greatest Common Factor

When factoring polynomials, the Greatest Common Factor (GCF) is a key concept that can simplify the expression. The GCF is the largest factor shared by all terms in a polynomial. Identifying the GCF is an important first step in making factorization easier.
Here’s how to spot and pull out the GCF:

• Look at each term in the polynomial and find the common factors they share.
• In this specific exercise, consider the terms inside each group. For instance, in the group \(x^4 + 3x^3\), \(x^3\) is the GCF because it is the highest power of \(x\) present in both terms.
• For the group \(-2x - 6\), the common factor is \(-2\), since both terms are divisible by -2.

Extract the GCF from each group to simplify the expression step by step. By reducing the polynomial into smaller parts using the GCF, further factoring methods, like grouping and special patterns, become more straightforward.

Grouping Technique

The Grouping Technique is a method for factoring polynomials that involves separating them into smaller groups and factoring each part separately. This method is especially useful when dealing with polynomials with four or more terms.
Steps to apply the Grouping Technique:

• Divide the polynomial into pairs or groups that may have common factors. In our case, the polynomial \(x^{4}+3x^{3}-2x-6\) was grouped as \((x^4 + 3x^3)\) and \((-2x - 6)\).
• Factor out the GCF from each group as learned previously.
• After factoring out, see if the resulting groups share a common binomial factor.

In our example, after factoring, both groups contained the factor \((x+3)\), which could be pulled out, simplifying the polynomial into a simpler binomial multiplication. Grouping effectively simplifies complicated polynomials and makes them easier to handle.

Difference of Cubes

A polynomial can sometimes be reduced further by recognizing special patterns, such as the Difference of Cubes. The Difference of Cubes formula helps to factor expressions of the form \(a^3 - b^3\). The formula is:\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]To use this in practice:

• Identify cubic terms: In \(x^3 - 2\), you spot \(x^3\) and consider \(2\) as \(\sqrt[3]{2^3}\).
• Apply the formula: Here \(a = x\), and \(b = \sqrt[3]{2}\).
• Substitute \(a\) and \(b\) into the formula to get: \((x-\sqrt[3]{2})(x^2 + x\sqrt[3]{2} + (\sqrt[3]{2})^2)\).

Learning and applying the Difference of Cubes can unveil the final factors of a polynomial, bringing us to a complete factorization. Understanding and practicing these formulas allows you to break down even the most complex polynomial expressions.