Problem 13 Multiply. $$6 v^{3}\left(v^{2}... [FREE SOLUTION]

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Chapter 6: Problem 13

Multiply. $$6 v^{3}\left(v^{2}-4 v-2\right)$$

Short Answer

Expert verified

The short answer is: $6v^5 - 24v^4 - 12v^3$.

Step by step solution

Identify the Monomial (Outer) and Polynomial (Inside Parentheses) Expressions

Here, the monomial expression is $6v^3$, and the polynomial expression inside the parentheses is $v^2 - 4v - 2$.

Distribute the Monomial Expression to Each Term Inside the Parentheses

Multiply the monomial $6v^3$ by each term inside the parentheses, one at a time: $6v^3 * v^2 = 6v^5$ $6v^3 * -4v = -24v^4$ $6v^3 * -2 = -12v^3$

Write the Resulting Expression

Combine the results obtained in Step 2: $6v^5 - 24v^4 - 12v^3$ So, the final result is $6v^5 - 24v^4 - 12v^3$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomial

In mathematics, a monomial is a simple algebraic expression that consists of one term. This term may have a constant, a variable, or both combined with non-negative integer exponents. Monomials like $6v^3$ have only one piece, making them straightforward to identify. Here's what a monomial entails:

**Coefficient**: The number part of the term. In $6v^3$, 6 is the coefficient.
**Variable**: The letter part that can change. Here, $v$ is the variable.
**Exponent**: The power that the variable is raised to. In this case, 3 is the exponent of $v$.

To understand monomials better, imagine each part as a building block. The coefficient indicates the size, the variable is the type, and the exponent shows how many times the type appears as a factor. Recognizing this structure helps in operations like multiplication, as seen in the original exercise.

Distributive Property

The distributive property is a fundamental algebraic concept used to multiply a single term and a sum or difference. It's the method of distributing a common factor across all elements inside a parenthesis. For instance, in the expression $6v^3(v^2 - 4v - 2)$, the distributive property lets us multiply each term inside the parentheses by $6v^3$. Here's how it works:

Multiply $6v^3$ with $v^2$ to get $6v^5$.
Multiply $6v^3$ with $-4v$ to get $-24v^4$.
Multiply $6v^3$ with $-2$ to get $-12v^3$.

This step-by-step approach shows how one distributes the outer monomial across each term inside the polynomial, ensuring every term is accounted for in the final expression. The distributive property simplifies complex expressions, making them easier to solve.

Exponent Rules

Exponent rules are essential in simplifying expressions involving powers, especially during multiplication. When multiplying terms like the ones in the exercise, understanding these rules is crucial. Here are key rules that apply:

Product of Powers Rule ($a^m \cdot a^n = a^{m+n}$): To multiply expressions with the same base, add their exponents. For example, $v^3 \cdot v^2 = v^{3+2} = v^5$.
Power of a Product Rule: Distribute the exponent to each factor within a monomial. If needed, apply it when products are raised to subsequent powers.

These rules make it clear how individual operations within an expression are performed, ensuring accuracy in simplifying terms like $6v^3 \cdot v^2$ without confusion. Mastering these exponent principles aids in handling more intricate algebraic expressions efficiently.

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Multiply. $$6 v^{3}\left(v^{2}-4 v-2\right)$$

Short Answer

Step by step solution

Identify the Monomial (Outer) and Polynomial (Inside Parentheses) Expressions

Distribute the Monomial Expression to Each Term Inside the Parentheses

Write the Resulting Expression

Key Concepts

Monomial

Distributive Property

Exponent Rules

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