Problem 24 - Technical Mathematics with Calculus

Question

$$5 a^{3} x^{2}-5 a^{2} x^{3}+10 a^{2} x^{2} z$$

Step-by-Step Solution

Answer

5a^3x^2 - 5a^2x^3 + 10a^2x^2z = 5a^2x^2(a - x + 2z)

Step 1: Identify common factors

Examine the terms to identify any common factors. Here, all three terms have a '5', 'a' to at least the second power, and 'x' to at least the second power.

Step 2: Factor out the common factors

Factor out the common factors from each term which is the greatest power of each that is present in all terms. The greatest common factor is 5, 'a^2' and 'x^2'. So we factor out 5a^2x^2 from each term.

Step 3: Rewrite the expression

After factoring out the common factors, rewrite the expression as the product of the common factors and the remainder terms. This gives us: 5a^2x^2(a - x + 2z).

Step 4: Simplify if necessary

Check if the remaining expression within the parentheses can be simplified. In this case, it cannot, so the simplified expression is: 5a^2x^2(a - x + 2z).

Key Concepts

Common Factors

When we talk about common factors, we're looking for numbers or variables that are shared by all terms within a polynomial. Just like finding common interests among friends, identifying common factors in algebra involves spotting the shared elements. In the polynomial 5a^3x^2 - 5a^2x^3 + 10a^2x^2z, each term includes the number 5, a variable 'a' raised to at least the second power, and the variable 'x' also raised to at least the second power. Think of this commonality as a sort of 'mathematical glue' that binds the terms together.

By extracting these factors, we're essentially pulling out the shared 'glue,' which simplifies the expression and makes our algebraic life a lot easier. Remember, the greatest common factor is the highest exponent of shared variables in all terms, along with any numerical coefficient they all have. Once identified, factoring polynomials becomes a much more approachable task.

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and operations combined to describe a particular relationship. Our example 5a^3x^2 - 5a^2x^3 + 10a^2x^2z is just that—a representation of a certain quantity where 'a', 'x', and 'z' can vary. It's a bit like a recipe with various ingredients mixed together in different amounts to create a unique dish. In algebra, the 'ingredients' are the 'terms' and they may comprise constants like 5, variables like 'a' and 'x', and exponents that tell you how much of a 'spice' to add.

To work effectively with algebraic expressions, becoming fluent in combining like terms and recognizing patterns is essential. This fluency helps in solving equations, simplifying expressions, and, of course, factoring polynomials efficiently which is intrinsic to algebraic manipulation.

Simplifying Expressions

The process of simplifying expressions is a lot like decluttering a room, where you organize and streamline items to make the space more functional and pleasing. In algebra, simplifying makes an expression easier to work with by reducing its complexity. After factoring out common factors from our polynomial, we're left with an expression that appears less complicated: 5a^2x^2(a - x + 2z). This particular expression couldn't be simplified further, but knowing when and how to simplify is a skill that applies to more than just polynomials.

Whether you're combining like terms, reducing fractions to their lowest terms, or performing arithmetic operations, the goal is always to pare down to the simplest form possible. Simplification is key to calculation efficiency and to understanding the underlying structure of algebraic expressions. Keeping an expression as straightforward as possible allows for clarity in problem-solving and communication within the language of mathematics.