Problem 59 - Introductory Algebra for College Students

Question

Factor completely. $$x^{4}-3 x^{3}-10 x^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial is \(x^{2}(x - 5)(x+2)\)

Step 1 - Common Factor Identification

First, analyze the equation to see if there is any common factor across the three terms. In this equation \(x^{4}-3 x^{3}-10 x^{2}\), \(x^{2}\) is a common factor.

Step 2 - Factor out Common Factor

Take out the common factor from the polynomial: \(x^{2}(x^{2} - 3x - 10)\)

Step 3 - Factor the quadratic

Now you have a quadratic \(x^{2} - 3x - 10\). Factor this quadratic as \(x^{2} - 5x + 2x - 10 = 0\).

Step 4 - Factor by Grouping

Group the factored equation into two pairs and factor each pair separately \(x^{2} - 5x + 2x - 10 = x(x-5) + 2(x-5)\)

Step 5 - Factor Full Equation

Combine the factored pairs to write the final factored equation: \(x^{2}(x - 5) + 2(x - 5) = (x^{2}+2)(x - 5)\)

Key Concepts

Common Factor Identification

Understanding how to identify common factors is crucial in simplifying polynomials. Suppose you're faced with the polynomial x^{4}-3x^{3}-10x^{2}. The first step is to look for a term that is present in each component of the polynomial. In our case, this term is x^{2}, which is a common factor for all three terms.

Extracting this common factor simplifies the polynomial significantly and lays the groundwork for further factoring. It's like untangling a knot by first loosening the easiest loop. By taking out x^{2}, we are left with x^{2}(x^{2}-3x-10), which is more manageable to work with. Always remember, if you skip this step, you might miss out on the simplest path to the solution.

Factoring by Grouping

Grouping is a technique that can be quite handy when dealing with polynomials that have four or more terms. After identifying and factoring out any common factors, the remaining polynomial may sometimes be broken down into smaller groups that can be factored further. It often involves rearranging the terms to find pairs that share a common factor.

Let's take our simplified polynomial x^{2}-3x-10. We can start by splitting the middle term to facilitate grouping: x^{2}-5x+2x-10. Now, we notice there are two pairs where a common factor is evident: x(x-5) and 2(x-5). This process effectively reveals a factor common to both groups, namely (x-5). Through this method, you have untied another knot in the equation, leaving you with an even simpler expression to deal with.

Factoring Quadratics

Quadratics are polynomials of the second degree, typically taking the form ax^{2} + bx + c. Factoring quadratics can sometimes be quite intuitive, as it often involves finding two numbers that both add up to b and multiply to ac. When we factored out x^{2} from our original polynomial, we were left with a quadratic: x^{2}-3x-10.

After rearranging the middle term into x^{2}-5x+2x-10 and applying grouping, we found that (x-5) was a factor. This gives us the factors of the quadratic as (x-5) and (x+2). When combined with the previously factored out x^{2}, we have fully factored the original polynomial. A solid understanding of how to factor quadratics is essential, as it is the foundation for solving higher-degree polynomial equations.