Problem 10 - Algebra 1: Concepts and Skills

Question

$$ 2 x\left(x^{2}+x-5\right) $$

Step-by-Step Solution

Answer

The expanded form of the given expression is \(2x^{3} + 2x^{2} - 10x\).

Step 1: Isolate terms within the parenthesis

Identify and isolate the terms within the parenthesis, which are \(x^{2}\), \(x\), and \(-5\) in this case.

Step 2: Apply Distributive Property

Apply the distributive property of multiplication over addition or subtraction by multiplying each term inside the parenthesis with 2x. This would give: \(2x*(x^{2}) + 2x*(x) + 2x*(-5)\).

Step 3: Simplify the terms

Simplify the terms from step 2 to obtain the expanded form of the expression: \(2x^{3} + 2x^{2} - 10x\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that involves the multiplication of a single term by each term within a set of parentheses. It's like distributing the outside term across the terms inside the parentheses. Here's how it works:

• Identify the term outside the parenthesis. In the given exercise, the term outside is \(2x\).
• Multiply this term by each individual term inside the parenthesis.

This means you take \(2x\) and multiply it by each of the three terms inside: \(x^2\), \(x\), and \(-5\). So, your operation will look like this: \(2x \cdot x^2 + 2x \cdot x + 2x \cdot (-5)\). This step is crucial as it sets up the equation for further simplification. By applying this property, you ensure that no term within the parentheses is left out in the final polynomial expansion.

Polynomial Simplification

Once the distributive property has been applied, the next step is polynomial simplification. Polynomial simplification involves combining any like terms and arranging the expression in the simplest form possible.

• First, perform all multiplicative operations as separated by the distributive step: \(2x \cdot x^2 = 2x^3\), \(2x \cdot x = 2x^2\), and \(2x \cdot (-5) = -10x\).
• Then, rewrite the expression with these new terms: \(2x^3 + 2x^2 - 10x\).

Each term results from multiplying \(2x\) with every term from the original parenthesis. This simplified expression is now fully expanded and does not have any like terms that can be combined further. Polynomial simplification helps in making algebraic expressions more manageable and easy to work with, particularly useful for further calculations or when solving equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. The exercise presented is a perfect example of an algebraic expression, showcasing how numbers and variables interact.

• The expression involves multiplying variables raised to a power (e.g., \(x^2\)) and constant terms (such as \(-5\)).
• Algebraic expressions do not have equal signs, differentiating them from algebraic equations.

Understanding algebraic expressions is crucial for solving complex equations and word problems in mathematics. They allow us to express relationships between variables and constants, helping in modeling real-world phenomena. In polynomial expansions like the exercise given, these expressions are transformed to reveal more concise forms, making it easier to interpret or solve problems.