Problem 24 - Algebra for College Students

Question

Find each product. $$\left(-7 x^{2}\right)(3 x)\left(4 x^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-84x^6\).

Step 1: Multiply the Coefficients

First, identify the coefficients in the expression: -7, 3, and 4. Multiply these coefficients together: \(-7 \times 3 \times 4 = -84\).

Step 2: Multiply the Variables

Next, identify and multiply the variable parts. Each term has a base of \(x\): the first term \(x^2\), second term \(x^1\), and third term \(x^3\). Add the exponents together: \(2 + 1 + 3 = 6\).This gives us \(x^6\).

Step 3: Combine the Results

Combine the result of multiplying the coefficients and the variable parts: The final expression is \(-84x^6\).

Key Concepts

Coefficients Multiplication

When multiplying polynomials, it is important to first identify and multiply the coefficients. Coefficients are the numerical parts of the terms. In the given expression, the coefficients are

• -7
• 3
• 4

To find the product of these coefficients, simply multiply them together: \(-7 \times 3 \times 4\). Calculating this step-by-step: - Multiply \(-7\) by \(3\) to get \(-21\).- Then multiply \(-21\) by \(4\) to find \(-84\).The result combines into the single coefficient of the final product. The multiplication of coefficients simplifies the expression's numerical factor in polynomial multiplication. Once you have the product of the coefficients, move on to the variable parts, which we'll explain next.

Exponents Addition

In polynomial multiplication, it's crucial to handle the variable parts by focusing on their exponents. An exponent indicates how many times a base is multiplied by itself. In our example, the base is \(x\). The expression contains three terms with these exponents:

• \(x^2\)
• \(x^1\)
• \(x^3\)

When multiplying variables with the same base, you add the exponents. Start by listing the exponents: 2, 1, and 3. Simply add them together: \(2 + 1 + 3 = 6\). This result, \(x^6\), represents the degree of the variable in the final expression.By adding the exponents, you efficiently combine the variable parts, ensuring the polynomial remains in a simplified form.

Algebraic Expressions

An algebraic expression consists of numbers and variables combined through operations like addition, subtraction, multiplication, and division. In this context, we are dealing with the multiplication of parts of an algebraic expression.Polynomials are a specific type of algebraic expression, made up of terms consisting of coefficients and variables raised to powers. Understanding how these parts interact when multiplied simplifies the process greatly. In our example, the algebraic expression is \((-7x^2)(3x)(4x^3)\). Working through this step-by-step:- Multiply the coefficients: Calculated as \(-84\).- Handle the variables by adding the exponents: Resulting in \(x^6\).- Combine: The final expression \(-84x^6\) effectively synthesizes the expression into a single, simplified algebraic expression.Successfully managing the parts of algebraic expressions is key to mastering polynomial multiplication and many other algebraic operations.