Problem 124 - Introductory Algebra for College Students

Question

Factor each polynomial. $$x^{2}-y^{2}+3 x+3 y$$

Step-by-Step Solution

Answer

The factorized form of the expression \(x^{2}-y^{2}+3 x+3 y\) is \(x(x+3)- y(y-3)\).

Step 1: Identify and factor out common terms

First, look at \(x^{2}+3 x\) and \(-y^{2}+3 y\). It can be seen that \(x\) is common to the terms in \(x^{2}+3 x\) and \(y\) is common to \(-y^{2}+3 y\). This allows us to factor out \(x\) from the first group and \(y\) from the second group.

Step 2: Apply the factoring

Apply the factoring to the grouped terms. This gives us \(x(x+3)- y(y-3)\). Notice that \(x+3\) and \(y-3\) are both binomials.

Step 3: Rearrange as desired

Polynomials can often be rearranged without changing their mathematical meaning. In this case, our polynomial can be left as \(x(x+3)- y(y-3)\) since it is fully factored. However, it may also be presented as \(x*(x+3) - y*(y-3)\), if desired.

Key Concepts

Polynomial Expression

Polynomials are algebraic expressions that consist of variables, coefficients, and constants combined together using addition, subtraction, and multiplication operations. A polynomial can have multiple terms, and each term is constructed from the product of a constant and one or more variables. The largest exponent in a polynomial determines its degree. For example, in the expression \( x^2 + 3x - y^2 + 3y \), there are four terms:

• \( x^2 \)
• \( 3x \)
• \(-y^2 \)
• \(3y \)

Each term has a degree, and in this example, the polynomial is of degree 2 because the term \( x^2 \) has the highest exponent. Polynomials can be manipulated through different operations including factoring, which is the process of breaking the polynomial into simpler components.

Binomial

A binomial is a specific type of polynomial that contains exactly two terms. For example, expressions such as \( x + 3 \) or \( y - 3 \) are binomials. These two-term expressions can be part of larger polynomial equations and are very common in algebra.When factoring polynomials, we often look to identify binomials that share a common factor or can be simplified further. For example, in the step-by-step solution of \( x^2 - y^2 + 3x + 3y \), two binomials \( x + 3 \) and \( y - 3 \) emerged after factoring out the common factors and grouping terms. Recognizing these simple forms within more complex expressions is crucial as it helps in simplifying and solving polynomial equations.

Common Factor

In algebra, identifying the common factor is a vital part of simplifying expressions and solving equations. A common factor is the expression or value that divides each of the components of a polynomial expression evenly. In the provided problem \( x^2 - y^2 + 3x + 3y \), factoring involved first identifying common factors within groups of terms:

• In \( x^2 + 3x \), the common factor is \( x \).
• In \(-y^2 + 3y \), the common factor is \( y \).

Pulling out these common factors helps in rewriting the expression in a simplified manner, allowing for easier manipulation and solving. Successfully factored expressions are often simpler to work with and can make solving equations more straightforward.

Algebraic Grouping

Algebraic grouping is a technique used to simplify complex polynomial expressions by arranging terms in a way that common factors can be identified and factored out. This method is particularly useful when dealing with polynomials that initially do not appear to be factorable.In the original exercise, the polynomial \( x^2 - y^2 + 3x + 3y \) was broken down into two manageable parts: \( x^2 + 3x \) and \(-y^2 + 3y \). By grouping terms strategically, we were able to factor out the common elements within each group, ultimately simplifying the expression to \( x(x+3) - y(y-3) \).Grouping facilitates the recognition of patterns such as binomials and enables the factoring process to take place more efficiently. It transforms a seemingly complex polynomial into a series of simpler expressions that are easier to handle in algebraic manipulations.