Problem 49 - Elementary Algebra

Question

Factor out the GCF. $$ 12 x 5 y 2-8 x 3 y $$

Step-by-Step Solution

Answer

The GCF is \(4x^3y\), so we factor to get \(4x^3y(3x^2y - 2)\).

Step 1: Identify the GCF of the Numerical Coefficients

Look at the coefficients of the terms, which are 12 and 8. The greatest common factor (GCF) of 12 and 8 is 4.

Step 2: Identify the GCF of the Variables

Examine the variables in the terms. Both terms have the variables \(x\) and \(y\). The lowest power for \(x\) in both terms is \(x^3\) and for \(y\) is \(y^1\). Thus, the GCF for the variables is \(x^3 y\).

Step 3: Factor Out the GCF from Each Term

The GCF of the entire expression is \(4x^3y\). Divide each term by \(4x^3y\) and write the expression as a product: \[12x^5y^2 - 8x^3y = 4x^3y(3x^2y - 2)\].

Step 4: Check the Factored Expression

Verify the factored form by distributing \(4x^3y\) back into \(3x^2y - 2\) to ensure you retrieve the original expression: \[4x^3y(3x^2y - 2) = 12x^5y^2 - 8x^3y\], confirming the factorization is correct.

Key Concepts

Factoring

Factoring is a crucial skill in algebra. Essentially, it means expressing a complex expression as a product of simpler expressions.
It’s like breaking down a big problem into smaller, more manageable parts. This helps to simplify expressions and solve equations.
When factoring, we look for common elements, like numbers and variables, shared by all the terms. This commonality is termed the Greatest Common Factor (GCF).

• The GCF for numbers is the largest number that divides each of the coefficients exactly.
• For variables, it is the variable raised to the lowest power present in each term.

Think of factoring as the reverse of expanding a product. Instead of multiplying, you're dividing out the common elements.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations.
They can represent a multitude of things, from simple equations to complex polynomials.

• An algebraic expression might involve addition, subtraction, multiplication, or division.
• They can also include variables raised to various powers.

When working with algebraic expressions, it is often necessary to simplify or manipulate them to solve for unknown values. This is where factoring comes in handy.
Factoring simplifies the expression and reveals underlying relationships between terms, making it easier to work with.

Variables

Variables serve as placeholders for numbers in algebra. They are represented by letters such as \(x\), \(y\), or \(z\).
These symbols can vary, which is why they're called "variables." In more complex expressions, multiple variables can be raised to different powers or combined in various ways. For instance, in \(12x^5y^2 - 8x^3y\):

• The variable \(x\) is raised to the power of 5 in the first term and 3 in the second term.
• The variable \(y\) is to the power of 2 in the first term and 1 in the second term.

Understanding how to work with variables is pivotal because they allow us to create general formulas and solve a wide range of problems. When factoring, we look for the smallest power of each variable present in all terms to find the GCF.