Problem 22 Find the product. $$ 4 w^{2}... [FREE SOLUTION]

Chapter 10: Problem 22

Find the product. $$ 4 w^{2}\left(3 w^{3}-2 w^{2}-w\right) $$

Short Answer

Expert verified

The product is $12w^{5} - 8w^{4} - 4w^{3}$

Step by step solution

Distribute the Term

First step is to distribute $4w^{2}$ to each of the terms inside the brackets. The product of $4w^{2}$ and $3w^{3}$ is $12w^{5}$, the product of $4w^{2}$ and $-2w^{2}$ is $-8w^{4}$ and the product of $4w^{2}$ and $-w$ is $-4w^{3}$. Thus the resulting expression is $12w^{5} - 8w^{4} - 4w^{3}$.

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Key Concepts

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

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions. It essentially enables us to distribute a single term across a series of terms inside parentheses.
In algebraic terms, the distributive property states that for any numbers or variables, $ a(b + c) = ab + ac $. This means that we multiply the term outside of the parentheses by each term inside the parentheses separately.

In our exercise, the term $4w^2$ is distributed to each term within the parentheses: $3w^3$, $-2w^2$, and $-w$.
The distributive property helps to eliminate the parentheses and reduce the complexity of the expression, simplifying it to a single polynomial that can be more easily managed or solved further.

By applying this property, we turn a complex product into a sum, making it easier to handle and interpret mathematical expressions.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. These expressions are used in algebra to represent relationships between numbers.
For example, $4w^2(3w^3 - 2w^2 - w)$ is an algebraic expression involving the variable $w$.

Algebraic expressions can include one or more terms added together, like $3w^3$ or $-2w^2$.
Expressions can be simplified or manipulated using various algebraic techniques, such as factoring, expanding, or distributing, as we see in this exercise.

Expressions form the building blocks of equations and inequalities, serving as essential tools in formulating and solving real-world problems.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In algebra, they are crucial for representing powers of variables.
The expression $w^3$ symbolizes $w$ multiplied by itself three times $(w \times w \times w)$.

When multiplying terms with the same base, exponents are added together. For example, $w^2 \times w^3 = w^{2+3} = w^5$.
In the exercise, we multiply terms with exponents and adjust the exponents according to the rule mentioned above.

Understanding exponents is crucial for handling many algebraic operations, simplifying expressions, and solving polynomial equations. By mastering the basics of exponents, students can better navigate complex algebraic processes.

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91影视

Find the product. $$ 4 w^{2}\left(3 w^{3}-2 w^{2}-w\right) $$

Short Answer

Step by step solution

Distribute the Term

Key Concepts

Distributive Property

Algebraic Expression

Exponents

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