Problem 15 - Elementary and Intermediate Algebra

Question

Multiply. See Example 1. $$ \left(3 x^{2}\right)\left(4 x^{3}\right) $$

Step-by-Step Solution

Answer

The product is \(12x^5\).

Step 1: Identify the Components

The expression to be multiplied is \((3x^2)\) and \((4x^3)\). We have two monomials here, each with a coefficient (3 and 4 respectively) and a variable part with exponents \(x^2\) and \(x^3\).

Step 2: Multiply the Coefficients

First, multiply the numerical coefficients of the two terms: 3 and 4. \[3 \times 4 = 12\].

Step 3: Add the Exponents for Like Bases

Next, multiply the variable parts with the same base \(x\). When multiplying powers with the same base, you add the exponents.\[x^2 \times x^3 = x^{2+3} = x^5\].

Step 4: Combine Results

Combine the results of the previous steps. Multiply the coefficients obtained in Step 2 with the variable part calculated in Step 3.This gives you:\[12x^5\].

Key Concepts

Understanding Coefficients in Monomials

In any polynomial expression, it's important to recognize coefficients, which are the numerical factor in front of variables. In our exercise, the monomials \((3x^2)\) and \((4x^3)\) have coefficients of 3 and 4 respectively.
When we multiply these monomials, the coefficients play a significant role. You simply multiply them together:

• Take the first coefficient, 3.
• Find the second coefficient, 4.
• Multiply them: \(3 \times 4 = 12\).

This step is crucial because it forms the numerical part of the final answer. Consequently, understanding how to work with coefficients is fundamental to polynomial multiplication.

The Role of Exponents in Polynomial Multiplication

Exponents are the small numbers written at the top right of a base number, showing how many times the base is multiplied by itself.
In our example, the exponents are 2 and 3 for the terms \(3x^2\) and \(4x^3\) respectively.
When multiplying expressions that have the same base, we focus on adding these exponents:

• The base here is \(x\).
• We have \(x^2\) and \(x^3\).
• Add the exponents: \(2 + 3 = 5\).

This gives us the new exponent in the multiplied term, resulting in \(x^5\).
Understanding this concept ensures that we correctly simplify expressions during multiplication.

Monomials: Building Blocks of Polynomials

Monomials are expressions with a single term which can include numbers, variables, and exponents. These are the most basic unit of polynomials and are crucial in algebra.
In the expression to be multiplied, we have two monomials: \((3x^2)\) and \((4x^3)\). Each consists of:

• A coefficient (3 and 4, respectively).
• A variable base (\(x\)).
• An exponent (2 in \(x^2\) and 3 in \(x^3\)).

When you multiply monomials:

• Multiply the coefficients and add the exponents of the same base.
• This leads to a single combined expression, as seen in the product \(12x^5\).

Mastering polynomial multiplication begins with understanding the structure and multiplication of monomials.