Problem 61 - Algebra A Combined Function

Question

Factor each polynomial by grouping. $$ \begin{aligned} &x^{2 n}+2 x^{n}+3 x^{n}+6\\\ &\text { (Hint: Don't forget that } \left.x^{2 n}=x^{n} \cdot x^{n} .\right) \end{aligned} $$

Step-by-Step Solution

Answer

The polynomial factors to \((x^n + 2)(x^n + 3)\).

Step 1: Rewrite the Polynomial

Start by rearranging the terms of the polynomial \(x^{2n} + 2x^n + 3x^n + 6\) into two groups. We'll write it as \((x^{2n} + 2x^n) + (3x^n + 6)\).

Step 2: Factor the First Group

In the first group, \(x^{2n} + 2x^n\), notice that \(x^n\) is a common factor. Factor it out to get \(x^n(x^n + 2)\).

Step 3: Factor the Second Group

In the second group, \(3x^n + 6\), \(3\) is a common factor. Factor it out to get \(3(x^n + 2)\).

Step 4: Identify the Common Binomial Factor

Both groups have a common factor of \((x^n + 2)\). Factor \((x^n + 2)\) from the expression \(x^n(x^n + 2) + 3(x^n + 2)\) to get \((x^n + 2)(x^n + 3)\).

Step 5: Verify the Solution

Expand the factored form \((x^n + 2)(x^n + 3)\) to check if it matches the original polynomial. The expansion gives \(x^n(x^n + 3) + 2(x^n + 3) = x^{2n} + 3x^n + 2x^n + 6 = x^{2n} + 5x^n + 6\). Since this matches the original polynomial reordered, our factorization is correct.

Key Concepts

Grouping Method

The grouping method is a clever technique used to simplify the process of factoring polynomials by breaking them down into manageable parts. This strategy is particularly useful when dealing with polynomials that have four or more terms, as seen in our original exercise. The goal is to rearrange and group terms in a way that simplifies the expression.

Here’s how the method works:

• Step 1: Identify groups of terms that share a common factor.
• Step 2: Factor out the common factor from each group.
• Step 3: Check if the groups now have a common binomial factor.
• Step 4: Factor out the common binomial to simplify further.

By following these steps, you break the polynomial into simpler parts, which can reveal a hidden common factor that wasn’t initially obvious.

In our exercise, the polynomial was divided into \(x^{2n} + 2x^n\) and \(3x^n + 6\), each of which was further broken down to reveal a common binomial.

Binomial Factor

A binomial factor refers to an expression consisting of two terms, which can often appear when factoring polynomials. In algebra, a binomial may look like \((x^n + 2)\) or \((x^n + 3)\), representing a sum or difference. These play a vital role in factorization because they offer a simplified form that can be used to extract larger, overarching patterns within polynomials.

During our solution process, we identified that both groups factored separately could be expressed in terms of the binomial \((x^n + 2)\). This allowed us to leverage the binomial factorization to condense the polynomial into a more workable form.

Recognizing binomial factors is a crucial skill in polynomial factorization. It often enables the simplification of complex algebraic expressions. Practicing the identification and utilization of these factors enriches a student’s algebraic capabilities and enhances problem-solving efficiency.

Common Factor

Finding the common factor in algebraic expressions is a fundamental skill that aids in simplifying and solving equations. A common factor is a term or number that divides each term in an expression without a remainder. Recognizing and extracting this factor is the first step in many algebraic manipulations.

In our task, the first group \(x^{2n} + 2x^n\) shared a common factor of \x^n\, and the second group \(3x^n + 6\) had a common factor of \3\. Factoring these out helps to expose a structure that makes it easier to simplify the expression further, specifically, revealing common binomial factors.

Keep these tips in mind when looking for common factors:

• Look for variables or numbers that appear in each term.
• Factor out the greatest of these terms to simplify.
• Reassess the expression to uncover any additional simplifications.

Extracting common factors is like pulling back layers of complexity within polynomial expressions, revealing an easier pathway toward the solution.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition or multiplication). They form the foundation of algebra, enabling both simple and complex mathematical problems to be analyzed and solved. Understanding how to manipulate these expressions is key to mastering topics like polynomial factorization.

Polynomials, a type of algebraic expression, consist of terms connected by addition or subtraction, where each term is a product of an integer coefficient and a variable raised to a non-negative integer power. For example, \(x^{2n} + 5x^n + 6\) is a polynomial expression.

Factoring these algebraic expressions is an important process because it reveals the underlying structure of the equation, often simplifying it for further operations, such as solving for \x\. By breaking down a polynomial into its component parts, you gain a clearer view of its overall behavior and characteristics.

To succeed with algebraic expressions, it’s crucial to:

• Understand the basic operations and properties of exponents and coefficients.
• Recognize patterns, like common factors and binomials.
• Practice different factorization techniques to handle a variety of forms and problems.

These principles make algebraic problem-solving more intuitive and approachable, allowing students to tackle more advanced topics with confidence.