Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(3 u-2 u^{2}+6-u^{3}\)

Step-by-Step Solution

Answer

The factorized form of the given expression \(3 u-2 u^{2}+6-u^{3}\) is \((u+2)(-u^{2}+3)\).

Step 1: Arrange the terms in descending order

First, sort the polynomial in descending order by degree. The given polynomial is \(3 u-2 u^{2}+6-u^{3}\). Rearranging this results to \(-u^{3}-2u^{2}+3u+6\).

Step 2: Factor out the GCD

After ordering, examine each term to locate any common factors. In this case, there's no common factor across all the terms. Therefore, proceed to the next step.

Step 3: Factor the polynomial

Factorize the polynomial. This equation can be factorized by grouping. Group the polynomial as \((-u^{3}-2u^{2})+(3u+6)\) then factor out the common factors from each group separately. This gives \(-u^{2}(u+2)+3(u+2)\).

Step 4: Final factorization

Finally, notice that the terms in parentheses in the previous step are the same. Therefore, the expression can be factored further into \((u+2)(-u^{2}+3)\).

Key Concepts

Factor by Grouping

The technique of factor by grouping involves breaking down a polynomial into separate groups in such a way that every group has a common factor. This method is particularly useful when you're dealing with a polynomial that doesn't appear to have any factors common to all terms.

Let's consider the polynomial -u^3-2u^2+3u+6. We can split this into two groups: one containing the terms -u^3 and -2u^2, and the other containing 3u and 6. Within these groups, we look for a greatest common factor. The first group has -u^2, resulting in -u^2(u+2), and the second group has 3, giving us 3(u+2). Notice how both groups contain a (u+2).

This leads to the insight that we can now express the original polynomial as the product of (u+2) and another polynomial, -u^2+3. The outcome is more visible when the terms are arranged with this same factor present in each group, which we factored out as a common term. Ultimately, this method simplifies polynomials that at first glance may not seem factorable.

Descending Order of Polynomial

The descending order of a polynomial refers to arranging the terms of the polynomial so that the exponents on the variables decrease from left to right. This order is useful not only for visual clarity but also for identifying terms that might be grouped during the factorization process.

For example, the initial expression 3u - 2u^2 + 6 - u^3 becomes clearer when it's rearranged as -u^3 - 2u^2 + 3u + 6. Not only does this descending order help to quickly identify the degree of the polynomial (which is 3 in this case), but it also sets up the problem nicely for techniques like factoring by grouping. It's essential to start with the highest power term, and systematically proceed to the lowest to make sure none of the terms are overlooked in the factorization process.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest number that divides two or more numbers. In the context of polynomials, the GCD is the highest degree of a polynomial that divides all the terms without leaving a remainder.

When attempting to factorize polynomials, finding the GCD can significantly simplify the process by reducing the terms to their simplest form. In our example, the terms within each group, namely -u^3 - 2u^2 and 3u + 6, at first glance don't seem to have a common factor across all four terms. However, on grouping, you can determine and factor out the GCD of each subgroup, which in this case turns out to be -u^2 and 3, respectively. It's important to note that sometimes there may not be a GCD for all terms, but this doesn't impede grouping; you can still look for common factors within subgroups as a pathway to further factorization.