Problem 80 - Introductory Algebra for College Students

Question

Factor by grouping. $$x y-x+7 y-7$$

Step-by-Step Solution

Answer

The factored form of the given expression is \((x + 7)(y - 1)\).

Step 1: Group the terms

Group the terms to that common factors can be identified. The polynomial \(xy - x + 7y - 7\) can be separated into two groups like this: \( (xy - x) + (7y - 7) \).

Step 2: Factor out the common terms

Now factor out the common terms in each group. In the first group, \(x\) is common and in the second group, \(7\). So, the factored expression becomes: \(x(y - 1) + 7(y - 1)\).

Step 3: Final Factoring

Notice that \(y - 1\) is now a common term for both groups. So, group \(y - 1\) together and then factor it out from both terms. The final factored expression will be: \((x + 7)(y - 1)\).

Key Concepts

Polynomial Factoring

Factorization is a fundamental skill in algebra, especially when tackling polynomials. In essence, factoring a polynomial means breaking it down into simpler components that, when multiplied together, reproduce the original polynomial. Think of it as the reverse operation of expanding.
Polynomial factoring relies on finding factors common to all terms, such as numbers, variables, or expressions. With practice, recognizing patterns in polynomials becomes easier. For example, if you see a term repeated in a polynomial—like the \(y - 1\) in our exercise—you're on track to factor by grouping, which simplifies the polynomial into a product of two binomials.
A good starting point is to memorize key factorization patterns, such as difference of squares \(a^2 - b^2 = (a + b)(a - b)\) and perfect squares \(a^2 + 2ab + b^2 = (a + b)^2\). With these patterns in hand, you can approach polynomial factoring with more confidence.

Common Factors

Spotting common factors is a bit like playing a detective game with numbers and variables. You are hunting for any numbers, variables, or expressions that appear in each term of a polynomial.
When you find a common factor, like \(x\) in the first group and \(7\) in the second group of our example \(xy - x + 7y - 7\), you can 'pull' this factor out in front of a set of parentheses. Inside these parentheses, you're left with the remaining parts of each term—all neatly simplified.
This process not only makes the expression easier to understand but also prepares it for further algebraic manipulation, such as solving equations or simplifying expressions. As a critical step in algebra, mastering the identification and extraction of common factors is indispensable.

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and operations (additions, subtractions, multiplications, etc.) that represents a specific quantity. They can be simple, like \(2x + 3\), or complex, like the one in our exercise, \(xy - x + 7y - 7\).
Understanding the structure of algebraic expressions is crucial. They are typically divided into terms, which are the different parts separated by plus or minus signs. Each term can be a single number (called a coefficient), a variable (like \(x\) or \(y\)), or a combination of both.
In algebra, manipulating these expressions—such as expanding, factoring, or simplifying—allows us to solve equations and inequalities or analyze functions. The more comfortable you are with different types of expressions, the more you'll be able to unlock the power of algebra.

Introductory Algebra

Introductory algebra sets the stage for understanding more complex mathematical concepts. It's where we first encounter symbols and letters that represent numbers and learn how to manipulate mathematical statements to find unknown values.
This branch of mathematics helps us transition from basic arithmetic to a more generalized way of thinking about numbers and relationships between them. Algebraic techniques, like factoring by grouping, are tools we use to simplify and solve equations.
Mastering the basics, such as understanding variables, coefficients, and algebraic expressions, is essential. It ensures that as problems become more complicated, we have the foundation needed to tackle them head-on. Staying patient and practicing consistently will lead to a profound understanding of algebra and its applications.