Problem 42 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -2 t(12-t) $$

Step-by-Step Solution

Answer

The given expression \(-2t(12 - t)\) simplified using the distributive property is \(2t^2 - 24t\).

Step 1: Identify the components

The given expression is \(-2t(12 - t)\). Here, -2t is like 'a', while (12 - t) is like a sum of 'b' and 'c' where 'b' is 12 and 'c' is -t.

Step 2: Apply the Distributive Property - Part 1

First, multiply 'a' which is -2t with 'b' which is 12. This gives the product -24t.

Step 3: Apply the Distributive Property - Part 2

Next, multiply 'a' which is -2t with 'c' which is -t. This gives the product 2t^2.

Step 4: Combine the Results

Combine both the products -24t and 2t^2. Since they're not like terms, they can't be simplified further. Therefore, the expression \(-2t(12 - t)\) is equivalent to \(2t^2 - 24t\) when the distributive property is applied.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating them. In this context, these symbols often represent numbers and operations. The foundation of algebra involves understanding how to express real-world situations with equations and expressions, as well as how to solve these problems.

• It allows for the abstraction of mathematical concepts, turning problems into general equations.
• By solving these equations, algebra provides a way to predict and understand patterns and relationships.

The exercise provided illustrates how algebraic techniques can simplify expressions with variables. Here, variables like \(-2t\) are involved, representing a general number that may change. By using algebraic methods such as the distributive property, one can transform and solve these expressions.

Polynomials

When dealing with algebraic expressions, you often come across polynomials. Polynomials are expressions consisting of variables and coefficients, assembled using addition, subtraction, and multiplication, but not division by a variable.

• Each part of the polynomial is called a "term." The expression \(-2t(12 - t)\) becomes a polynomial when expanded to \(2t^2 - 24t\).
• Polynomial terms are generally ordered by their degree, with the highest degree term first.

In our example, the polynomial consists of two terms: \(2t^2\) and \(-24t\). The degree of the first term is 2, as it contains \(t\) squared, while the second term has a degree of 1. Polynomials are foundational in algebra, forming the basis for many rules and operations you will encounter.

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic expressions into their simplest form. Here is where the distributive property, a key algebraic property, comes into play.
By applying the distributive property, you're multiplying one term by each part of a sum or difference inside parentheses, effectively removing the parentheses. This principle helps in transforming \(-2t(12 - t)\) into \(2t^2 - 24t\).

• First, distribute: multiply \(-2t\) by both \(12\) and \(-t\) individually.
• Then, simplify by combining all like terms (if there are any).

The result is a simpler expression that provides clarity and makes further mathematical operations more manageable. Simplifying expressions is essential because it facilitates further computation and strengthens understanding of complex equations.