Problem 50 - Elementary Algebra

Question

For the following problems, list, if any should appear, the common factors in the expressions. $$ \frac{3}{4} x^{2} y^{2} z^{2}+\frac{3}{8} x^{2} z^{2} $$

Step-by-Step Solution

Answer

The common factors present in both terms of the given algebraic expression are \(\frac{3}{4}x^2z^2\).

Step 1: Identify the factors of each term

The given expression consists of two terms: \(\frac{3}{4}x^2y^2z^2\) and \(\frac{3}{8}x^2z^2\). Let's first identify the factors of each term separately. For the first term, \(\frac{3}{4}x^2y^2z^2\), the factors are: - Numerical factor: \(\frac{3}{4}\) - \(x^2\) (since there are 2 x's multiplied) - \(y^2\) (since there are 2 y's multiplied) - \(z^2\) (since there are 2 z's multiplied) For the second term, \(\frac{3}{8}x^2z^2\), the factors are: - Numerical factor: \(\frac{3}{8}\) - \(x^2\) (since there are 2 x's multiplied) - \(z^2\) (since there are 2 z's multiplied)

Step 2: Compare the factors and list the common factors

Now, let's compare these factors and determine any factors that are common to both terms. The common factors are: - Numerical factor: From the numerical factors, \(\frac{3}{4}\) and \(\frac{3}{8}\), the common factor is \(\frac{3}{4}\). - Variable factors: \(x^2\) and \(z^2\) are the common factors among the variable factors, as both terms have those factors. So, the common factors in the given expression are \(\frac{3}{4}x^2z^2\).

Key Concepts

Common Factors

When dealing with algebraic expressions, identifying common factors plays a key role in simplifying and factoring expressions effectively. Common factors are components that appear in each term of an expression. By breaking down each term into its individual factors, we can easily spot what they have in common.

In the expression given, we have two terms:

• \(\frac{3}{4}x^2y^2z^2\)
• \(\frac{3}{8}x^2z^2\)

Finding common factors involves comparing the numerical components and the algebraic variables:

• **Numerical Factors**: Between \(\frac{3}{4}\) and \(\frac{3}{8}\), the common numerical factor is \(\frac{3}{4}\).
• **Variable Factors**: Both terms have \(x^2\) and \(z^2\), making these the common variable factors.

Using the common factors helps streamline expressions, paving the way for further simplification or factorization tasks.

Algebraic Expressions

Algebraic expressions involve combinations of variables, numbers, and operations. They represent formulas or quantities in mathematical analysis, often used to express relationships between different variables.

An algebraic expression can include:

• **Constants**: Fixed numerical values like 3 or \(\frac{1}{2}\).
• **Variables**: Symbols such as \(x, y, z\) representing numbers.
• **Operands**: Operations like addition, subtraction, multiplication, or division connecting constants and variables.

In our example, \(\frac{3}{4}x^2y^2z^2 + \frac{3}{8}x^2z^2\), the expression showcases terms with similar variables and differing coefficients. Recognizing the structure of such expressions is essential for methods like factoring, simplifying, or solving equations.

Simplifying algebraic expressions by factoring out common factors can aid in solving equations more efficiently.

Polynomials

Polynomials are specific types of algebraic expressions that consist of terms composed of variables raised to whole number powers, multiplied by coefficients. They are a fundamental concept in algebra, used to describe a wide range of phenomena.

Each term in a polynomial is a product of a coefficient and variables elevated to a power. The term degree is determined by the highest power of the variable in the polynomial. Polynomials can be very simple, like \(2x + 3\), or more complex, like our given example.

In the expression \(\frac{3}{4}x^2y^2z^2 + \frac{3}{8}x^2z^2\), each part follows the polynomial structure:

• **Terms**: Each separate part like \(\frac{3}{4}x^2y^2z^2\).
• **Coefficients**: The numerical factor, such as \(\frac{3}{4}\) in the first term.
• **Exponents**: The powers of variables like \(x^2\) indicating the degree of each term.

Grasping polynomials' nature enables understanding other algebraic concepts, facilitating solving and simplifying expressions.