Problem 59 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$(8+2 f) g$$

Step-by-Step Solution

Answer

Question: Expand the expression using the distributive property: $$(8+2f)g$$ Answer: The expanded expression is $$8g + 2fg$$.

Step 1: Identify the terms inside the parentheses

Inside the parentheses, we have two terms: 8 and 2f.

Step 2: Apply the distributive property

To apply the distributive property, we will multiply both terms by g: $$g(8) + g(2f)$$

Step 3: Simplify the expression

Finally, we can simplify the expression by performing the multiplication: $$8g + 2fg$$ The expanded expression using the distributive property is: $$(8+2f)g = 8g + 2fg$$.

Key Concepts

Algebraic Expressions

At the heart of algebra, we often work with expressions that include numbers, variables, and operators. These are called algebraic expressions. An algebraic expression can be as simple as a number or variable, or it can be more complex, involving several numbers and variables combined by operations like addition, subtraction, multiplication, and division.
For example, in the expression \(8 + 2f\), '8' is a constant term, while '2f' is a term that involves the variable 'f' and a coefficient '2'. Each term in an algebraic expression is separated by an addition or subtraction sign.

• Terms: Parts of an expression separated by + or - signs.
• Coefficients: Numbers like '2' that multiply a variable.
• Variables: Symbols, like 'f', that can represent varying quantities.

Understanding these parts is crucial, as they form the foundation for manipulating expressions, including using the distributive property to expand them.

Polynomials

Polynomials are a specific class of algebraic expressions that include one or more terms. Each term in a polynomial is a product of a constant and a variable raised to a whole number exponent. When you encounter expressions like \(8 + 2f\), you are working with polynomials. In this specific case, it is a binomial since it consists of two terms.

• Monomial: A single-term polynomial, like \(3x\).
• Binomial: A two-term polynomial, like \(8 + 2f\).
• Trinomial: A three-term polynomial, such as \(x^2 + 3x + 2\).

Polynomials are fundamental because they allow us to create equations and functions that describe real-world situations. They also help in understanding relationships between variables, which is key in solving algebraic equations.

Expansion of Expressions

The expansion of expressions involves the process of removing parentheses in an expression and writing it out in an expanded form. This is where the distributive property becomes essential. The distributive property states that multiplying a term by a sum or difference inside parentheses is the same as multiplying each term separately.
For the expression \((8+2f)g\), applying the distributive property involves multiplying 'g' by each term inside the parentheses:

• Step 1: Multiply 'g' by '8', which results in \(8g\).
• Step 2: Multiply 'g' by '2f', resulting in \(2fg\).
• Step 3: Combine the results to get the expanded expression: \(8g + 2fg\).

Expanding expressions is crucial in algebra because it simplifies equations and makes them easier to work with, particularly when solving them.