Problem 84 - Introductory Algebra for College Students

Question

Factor completely. $$6 x^{2} y-2 x y-60 y$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial is \(2y (3x + 10)(x - 3)\).

Step 1: Identifying the Common Factor

The first step in factoring a polynomial is to identify the common factor among all terms. Here, all terms are multiples of \(2y\). So, \(2y\) is the common factor.

Step 2: Factoring Out the Common Factor

We can factor out the common factor by dividing each term by the common factor, \(2y\). Doing this, we get \(2y (3x^2 -x - 30)\).

Step 3: Factoring the Remaining Polynomial

Now, the remaining polynomial \(3x^2 -x - 30\) can also be factored further into \((3x + 10) (x - 3)\). Combining this with the common factor \(2y\), we get \(2y (3x + 10)(x - 3)\).

Key Concepts

Common Factor

Identifying a common factor is the first and crucial step in simplifying algebraic expressions, especially polynomials. A common factor is a number or expression that evenly divides all terms within a polynomial expression.
In the exercise provided, we have the polynomial expression:

• \(6x^2y - 2xy - 60y\)

By examining the expression, we see that each term shares a common factor, which is \(2y\).
Here's how you can tell:

• All coefficients are divisible by 2.
• All terms have at least one \(y\).

Factoring out \(2y\) simplifies the problem and sets the stage for more advanced factoring in later steps.

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial into simpler, more manageable factors. This process can go beyond finding a common factor; it includes expressing the polynomial as a product of two or more polynomials.
After extracting the common factor \(2y\), we are left with the polynomial:

• \(3x^2 - x - 30\)

This expression can be factored further using techniques such as splitting the middle term or applying the quadratic formula. The goal is to find two binomials that, when multiplied, give the original polynomial. For our case:

• \((3x + 10)\)
• \((x - 3)\)

This results in the complete factorized form:

• \(2y(3x + 10)(x - 3)\)

Polynomial factorization helps in solving equations, simplifying expressions, and understanding polynomial functions.

Algebraic Expressions

Algebraic expressions form the foundation of algebra, providing a way to express relationships and quantities systematically. An algebraic expression is a combination of numbers, variables, and operators (such as +, -, *, /). In the context of polynomial expressions, we deal specifically with terms involving powers of variables.
In the exercise at hand, the expression:

• \(6x^2y - 2xy - 60y\)

is an algebraic expression comprising three terms. It includes variables \(x\) and \(y\), and coefficients (such as 6, -2, and -60).
Recognizing such expressions and how to manipulate them, like factoring, is a key skill in algebra. It allows you to simplify complex problems, solve algebraic equations, and apply mathematical reasoning to real-world problems.