91Ó°ÊÓ

When multiplying polynomials, especially when variables are involved, understanding the multiplicative property of exponents is essential. This property helps us simplify expressions by organizing terms in an efficient way. The rule is: when you multiply two expressions with the same base, you add the exponents.
For example, consider multiplying \( n^a \) by \( n^b \). The result is \( n^{a+b} \). It is as simple as that.
Seeing this in action, when multiplying \( 2n \) by \( 7n^2 \), we have:

• The base is \( n \)
• The exponents are 1 (from \( n \)) and 2 (from \( n^2 \))
• Add the exponents: \( 1 + 2 = 3 \)

So, the outcome is \( 14n^3 \), a clean combination of coefficients (2 and 7) and the newly summed exponent.

The distributive property is a fundamental concept in algebra, involving the distribution of one expression over the terms of another. It is what enables us to "distribute" the multiplication across each term within parentheses. In our exercise, the distributive property helps us expand the expression \( 2n\left(7n^2-5n+3\right) \).
To utilize this property, follow this method:

• Multiply \(2n\) by each term inside the parentheses, one at a time.
• First, distribute \(2n\) to \(7n^2\), yielding \(14n^3\).
• Next, distribute \(2n\) to \(-5n\), resulting in \(-10n^2\).
• Finally, distribute \(2n\) to 3, which gives \(6n\).

This distribution process allows us to effectively "spread" the removing factor across all terms, simplifying the multiplication task.

Understanding algebraic expressions is vital to manipulating and solving them effectively. An algebraic expression is a mathematical phrase that includes numbers, variables, and performs operations like addition, subtraction, multiplication, and division. Importantly, they do not include an equals sign, unlike algebraic equations.
In our exercise, the expression \(2n(7n^2-5n+3)\) consists of a single number expression \(2n\) and a polynomial \((7n^2-5n+3)\). This polynomial contains multiple terms:

• \(7n^2\), a term with a degree of 2
• \(-5n\), a term with a degree of 1
• 3, a constant term with a degree of 0

Each of these terms needs to be understood individually before engaging them in a multiplication context like ours. Comprehending these components enables us to break down and solve complex expressions step-by-step efficiently.