Question

$$5 a^{2} b+6 a^{2} c$$

Short answer

The factored expression is \(a^2(5b + 6c)\).

Step-by-step solution

Identify Common Factors

Look for factors that are common to each term in the expression. In the given expression, the common factor to both terms is the variable part \(a^2\).

Factor Out the Common Variable Part

Since \(a^2\) is a common factor in both terms, factor it out to simplify the expression. This results in \(a^2(5b + 6c)\).

Key concepts

Understanding Common Factors

When we talk about common factors in the context of algebra, we're referring to any number, variable, or combination of both that is present in all terms of an algebraic expression. For instance, consider the polynomial expression 5a^2b + 6a^2c. Here, both terms contain a^2. This a^2 is a common factor.

To identify common factors, inspect each term of the expression and look for elements that repeat. In this case, we see that a is squared in both terms. So, why is factoring out common factors important? Doing so simplifies the expression and prepares it for further operations, whether that be addition, subtraction, multiplication, division, or even more advanced algebraic manipulations.

Factoring isn't only about making the expression look neater; it can reveal deeper relationships between the terms and often simplifies complex problems. This is why the first step in solving algebraic expressions frequently involves looking for and factoring out the common factors.

The Art of Simplifying Expressions

Simplifying algebraic expressions is much like cleaning up a cluttered room; it's about organizing and reducing the clutter to make it more understandable. To simplify an expression, combine like terms and use arithmetic operations to make the expression as straightforward as possible. Factoring, just as we've identified common factors, is one of the primary tools for simplifying.

Once you've found and factored out the common factors from the polynomial expression, the remaining parts of each term are often more easily managed. In the example 5a^2b + 6a^2c, factoring out a^2 simplifies the expression to a^2(5b + 6c). This is significantly neater and helps us to quickly see and understand the relationship between the parts of the expression.

A key to simplifying is to always perform one operation at a time and double-check for any additional common factors. This meticulous approach ensures you won't overlook anything that could further simplify the expression.

Algebraic Expressions and Their Components

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs, but will not have an equality sign as an equation would. Think of it like a sentence that tells you how to combine different mathematical 'words' or numbers. A simple example is 5a^2b + 6a^2c; it includes variables (a, b, c), coefficients (5, 6), and an operation (addition).

Each part of an algebraic expression tells us something about the quantity it represents. The coefficients (5 and 6) are numerical and indicate how many of the variable parts are to be included. Variables stand in place for unknowns or quantities that can vary, and exponents like the squared symbol (2) on 'a' tell us that the variable is multiplied by itself.

Understanding the different parts and functions within an algebraic expression is crucial because it affects how we manipulate and ultimately solve these mathematical problems. This foundational knowledge is your ally in tackling algebra and advancing into more complex areas of mathematics.