91Ó°ÊÓ

Monomials are the simplest type of polynomial. They consist of a single term, which is a product of a constant number and one or more variables raised to an exponent. For instance, in the expression \(2x\), '2' is the coefficient, and 'x' is a variable with an implicit exponent of 1. When dealing with more complex polynomials, recognizing monomials helps to simplify processes, especially in operations like multiplication. Understanding how monomials interact with each other is crucial for solving algebraic problems efficiently. In our exercise, \(2x\) itself is a monomial involved in multiplying another polynomial.

The distributive property is a fundamental idea in algebra that allows us to multiply a term over a bracketed expression. It states that multiplying a sum or difference by a number gives the same result as multiplying each addend individually by the number and then performing the addition or subtraction.

Following this property, in our problem, we distribute \(2x\) to each term inside the parentheses:

• Multiply \(2x\) by the first term \(4x^3\), resulting in \(8x^4\).
• Then multiply \(2x\) by the second term \(5x^2\), resulting in \(10x^3\).

Applying the distributive property helps break down complex problems into manageable steps, making it easier to handle calculations correctly.

Exponent rules are essential for manipulating terms when performing operations like multiplication. One of the key rules is the product of powers rule, which states that when you multiply two exponents with the same base, you add the exponents together: \(a^m \cdot a^n = a^{m+n}\).

In our exercise, applying this rule is essential. As you multiply \(2x\) with \(4x^3\), we see that the exponents from each 'x' get added: \(x^1 \cdot x^3 = x^{1+3} = x^4\). Similarly, for \(2x \cdot 5x^2\), the exponents of 'x' are added: \(x^1 \cdot x^2 = x^{1+2} = x^3\).

Understanding and consistently applying exponent rules ensure precise results, providing clarity and structure to the solution process.