Problem 11 - College Algebra Essentials

Question

$$\text { Factor by grouping.}$$ $$x^{3}-2 x^{2}+5 x-10$$

Step-by-Step Solution

Answer

The factored form of \(x^{3}-2 x^{2}+5 x-10\) by grouping is \((x-2)(x^{2}+5)\).

Step 1: Identify the Groups

Break the expression into two groups: \(x^{3}-2 x^{2}\) and \(5 x-10\). Grouping is often done by halving the number of terms, but one should always mind the signs.

Step 2: Factor Out Common Factors in Each Group

Factor out the greatest common factor from each group. From \(x^{3}-2 x^{2}\), factor out \(x^{2}\) to get \(x^{2}(x-2)\). From \(5 x-10\), factor out 5 to get \(5(x-2)\).

Step 3: Final Factoring

Notice that the expressions in the parentheses are the same, so factor out the common binomial \(x-2\) which gives us \((x-2)(x^{2}+5)\).

Key Concepts

Polynomial Expression

A polynomial expression is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's a collection of terms added together, where each term is a product of a constant and a variable raised to an exponent. The variable usually appears in alphabetical order and with whole number exponents.

Polynomials can take many forms, such as:

• Linear: a polynomial of degree 1, such as \(x + 1\).
• Quadratic: a polynomial of degree 2, like \(x^2 + 2x + 1\).
• Cubic: a polynomial of degree 3, such as \(x^3 + 3x^2 - 2x + 7\).

In this exercise, we're dealing with a cubic polynomial: \(x^3 - 2x^2 + 5x - 10\). Identifying the degree of a polynomial is vital as it helps in selecting the appropriate factoring techniques.

Factor by Grouping

Factor by grouping is a method used to simplify polynomial expressions, making it easier to solve or manipulate them. This technique is particularly useful when dealing with polynomials that have four or more terms. The main goal is to rearrange and group the terms so that they share a common factor within each group.

Let's explore how factor by grouping works:

• Create groups: The first step involves breaking down the expression into two smaller groups. For the polynomial \(x^3 - 2x^2 + 5x - 10\), consider splitting it into groups: \(x^3 - 2x^2\) and \(5x - 10\).
• Factor each group: Once grouped, the focus is to find the greatest common factor (GCF) of each group. For example, \(x^3 - 2x^2\) factors as \(x^2(x - 2)\) and \(5x - 10\) factors as \(5(x - 2)\).
• Combine: Finally, if each group shares a common binomial, it can be factored out, leading to the final answer. Here, \(x - 2\) is common, which results in \((x - 2)(x^2 + 5)\).

By arranging terms into groups, this method takes advantage of the distributive property to simplify polynomials.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by two or more terms. Finding the GCF is a core step in simplifying or factoring expressions.

To find the GCF, consider each term's coefficients and variables separately:

• For coefficients, it's the largest number that divides each of them. For instance, if the terms are 10 and 15, their GCF is 5.
• For variables, the GCF is determined by the smallest power of a common variable across all terms.

In the exercise, for the group \(x^3 - 2x^2\), the GCF is \(x^2\), considering both the variable \(x\) and its lowest exponent within the group.

For \(5x - 10\), the GCF is 5, which is the largest number that can evenly divide both coefficients. Identifying and factoring out the GCF is essential as it simplifies expressions and lays the foundation for more advanced operations like factoring by grouping.