Problem 28 - Algebra for College Students

Question

Factor completely. $$27 x y-36 y$$

Step-by-Step Solution

Answer

The expression factors to \(9y(3x - 4)\).

Step 1: Identify Common Factors

Begin by identifying the greatest common factor (GCF) of the terms in the expression \(27xy - 36y\). Both terms share a factor of \(9y\).

Step 2: Factor Out the GCF

Factor \(9y\) out of each term in the expression: \[ 27xy - 36y = 9y(3x - 4) \]When you divide each term by \(9y\), you're left with \(3x\) and \(-4\).

Step 3: Verify the Factored Expression

To ensure the expression is factored completely, distribute \(9y\) back through \(3x - 4\) to check that it results in the original expression:\[ 9y(3x - 4) = 27xy - 36y \]This confirms that the factoring is correct.

Key Concepts

greatest common factor (GCF)

The greatest common factor, or GCF, is the largest factor that divides two or more numbers or terms without leaving a remainder. It is an essential concept in factoring algebraic expressions.
When you have an expression like \(27xy - 36y\), identifying the GCF helps simplify the expression by grouping terms that share common factors.

Here's how you can identify the GCF:

• List the factors of each term. For instance, for \(27xy\), the factors are 1, 3, 9, 27, x, y; and for \(36y\), they are 1, 2, 3, 4, 6, 9, 12, 18, 36, y.
• Identify the common factors. For both \(27xy\) and \(36y\), the common factors include 1, 3, 9, and y.
• Choose the greatest factor from the common factors. Here, \(9y\) is the greatest common factor."
  

Recognizing the GCF allows you to factor expressions more efficiently, simplifying complex expressions into more manageable forms.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They form the foundation of algebra and come in many forms, such as polynomials, equations, and inequalities.
Understanding these expressions starts by recognizing the variables and coefficients involved. For example, in \(27xy - 36y\), "27" and "36" are coefficients, and "x" and "y" are variables.

Here's a breakdown of common elements in algebraic expressions:

• Terms: The individual components of an expression, like \(27xy\) and \(-36y\).
• Variables: Symbols that represent numbers. In this expression, x and y are variables.
• Coefficients: Numbers multiplying a variable or variables. Here, 27 is the coefficient of xy, and 36 is the coefficient of y.
• Constants: Numbers on their own, though not present in this specific example.

Understanding these components helps in tasks such as simplification and factoring, by providing clarity on how expressions are constructed.

polynomial factorization

Polynomial factorization involves breaking down a polynomial into its simplest building blocks, or factors. This process simplifies polynomials and reveals their underlying relationships.
For the expression \(27xy - 36y\), the goal is to express it as a product of its factors. The steps to achieve this involve identifying and factoring out the greatest common factor (GCF).

When factoring polynomials, follow these steps:

• Identify the GCF for all terms in the polynomial. Here, it is \(9y\).
• Divide each term by the GCF, resulting in a simplified expression. For \(27xy - 36y\), dividing by \(9y\) leaves \(3x - 4\).
• Express the polynomial as the product of the GCF and the simplified expression. This makes \(27xy - 36y = 9y(3x - 4)\).

The resulting factored form is often easier to work with, especially in solving equations or simplifying expressions. Polynomial factorization is an invaluable tool in algebra, resolving complex expressions into more intuitive components.