Problem 8 - Elementary Algebra

Question

Determine the following products. $$ 6.03\left(2.11 a^{3}+8.00 a^{2} b\right) $$

Step-by-Step Solution

Answer

Answer: The result of distributing 6.03 to the expression $(2.11 a^3+8.00 a^2 b)$ is $12.7233 a^3 + 48.24 a^2 b$.

Step 1: Distribute the constant 6.03 to each term inside the parentheses.

To distribute the constant 6.03, we will multiply it by each term inside the parentheses: $$ 6.03\left(2.11 a^3+8.00 a^2 b\right) = 6.03(2.11 a^3) + 6.03(8.00 a^2 b) $$

Step 2: Multiply the constants in each term.

Now we will multiply the constants in each term: $$ 6.03(2.11 a^3) + 6.03(8.00 a^2 b) = 12.7233 a^3 + 48.24 a^2 b $$

Step 3: Write the final result.

The final result after distribution and simplification is: $$ 6.03\left(2.11 a^3+8.00 a^2 b\right) = 12.7233 a^3 + 48.24 a^2 b $$

Key Concepts

Distributive Property

The distributive property is one of the most important concepts in algebra. It allows you to multiply a single term (often called a constant or coefficient) by each term inside a set of parentheses.
This property is expressed as: \( a(b + c) = ab + ac \). It simplifies expressions and makes it easier to solve equations.
In the original exercise, the constant 6.03 is distributed to each term inside the parentheses. It involves two steps:

• First, multiply 6.03 by \( 2.11a^3 \).
• Second, multiply 6.03 by \( 8.00a^2b \).

By distributing this way, you're ensuring that each part of the polynomial gets multiplied by the external factor.
This step-by-step distribution helps simplify the polynomial multiplication process.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two or more polynomials together. It often involves using the distributive property on more complex expressions to combine like terms efficiently.
Understanding polynomial multiplication is essential for simplifying expressions, solving equations, and transforming equations into more manageable forms.In the given exercise, the process starts by multiplying each term inside the parentheses \( (2.11a^3 + 8.00a^2b) \) with the external constant 6.03.
As you carefully apply the constant to each term, you handle:

• \( 6.03 \times 2.11a^3 \) resulting in \( 12.7233a^3 \)
• \( 6.03 \times 8.00a^2b \) resulting in \( 48.24a^2b \)

After applying multiplication across all terms, you achieve a simplified expression, which is an essential part of algebraic manipulation.

Constant Multiplication

Constant multiplication involves taking a constant number and multiplying it by each term of a polynomial or algebraic expression. It is a straightforward yet pivotal operation in algebra that helps in simplifying and scaling equations.
In the context of the exercise, constant multiplication is used to apply the number 6.03 to each term within the parentheses.
This involves two simple, yet crucial calculations:

• First, multiply \( 6.03 \) by \( 2.11a^3 \), producing \( 12.7233a^3 \).
• Second, multiply \( 6.03 \) by \( 8.00a^2b \), resulting in \( 48.24a^2b \).

These operations maintain the structure of the algebraic expression while adjusting its scale. The purpose is to apply consistent transformation across each term to reflect the influence of the original constant on the expression.