Problem 81 - Elementary Algebra

Question

\(6 z\left(z^{2}\right)+(6 z)(8 z)+6 z(4)\)

Step-by-Step Solution

Answer

6z^3 + 48z^2 + 24z

Step 1: Distribute the common factor

Factor out the common term, which is 6z, from each term in the expression. The given expression is: \[ 6z(z^2) + 6z(8z) + 6z(4) \]When distributing 6z, you get: \[ 6z \times z^2 + 6z \times 8z + 6z \times 4 \]

Step 2: Simplify each term

Multiply the common factor (6z) with each term inside the parentheses.For the first term: \[ 6z \times z^2 = 6z^3 \]For the second term: \[ 6z \times 8z = 48z^2 \]For the third term: \[ 6z \times 4 = 24z \]

Step 3: Combine the simplified terms

Write down the combined simplified terms to form the final expression: \[ 6z^3 + 48z^2 + 24z \]So, the simplified version of \[6z(z^2) + 6z(8z) + 6z(4)\] is \[6z^3 + 48z^2 + 24z\].

Key Concepts

Distribution

Distribution is a fundamental concept in algebra that involves multiplying each term within a parentheses by a common factor outside the parentheses. In the exercise given, the expression is
6z(z^2) + 6z(8z) + 6z(4). To distribute means to spread out the factor across each term inside the parentheses:

• 6z × z^2
• 6z × 8z
• 6z × 4

The goal is to simplify the expression by multiplying each term by the common factor (6z). This process helps in breaking down complex expressions into more manageable pieces.

Common Factor

A common factor is a term that can be factored out from all terms in an expression. In the exercise, the common factor is 6z. Factoring out 6z means you recognize that each term in the expression (6z(z^2) + 6z(8z) + 6z(4)) includes 6z. By doing so, you can simplify the expression:

• First term: 6z × z^2 = 6z^3
• Second term: 6z × 8z = 48z^2
• Third term: 6z × 4 = 24z

The common factor helps in rewriting complex polynomial expressions in a more simplified form.

Polynomial Multiplication

Polynomial multiplication involves multiplying polynomials together. In the steps outlined, polynomial multiplication occurs when you multiply the common factor (6z) by each term of the polynomial inside the parentheses (z^2, 8z, and 4).

- For the term 6z × z^2, you get 6z³, because z × z^2 = z^3
- For the term 6z × 8z, you get 48z², because z × 8z = 8z²
- For 6z × 4, you get 24z, because z × 4 = 4z

Using polynomial multiplication makes it easier to combine the resulting products into a single simplified polynomial.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain numbers, variables, and operation symbols. They represent quantities in math that may not initially have a single, determined value. In the problem given, the algebraic expression is 6z(z^2) + 6z(8z) + 6z(4).

By understanding:

• How to distribute terms
• Identifying common factors
• Using polynomial multiplication

You can simplify any complex algebraic expression effectively. Each algebraic expression can be further dissected and simplified using these core concepts to achieve the final simplified form, such as 6z^3 + 48z^2 + 24z.