Problem 1 - Algebra for College Students

Question

Find each product. $$\left(4 x^{3}\right)(9 x)$$

Step-by-Step Solution

Answer

The product is \(36x^4\).

Step 1: Identify the Expressions

We have two expressions that need to be multiplied: \(4x^3\) and \(9x\). Each expression contains numerical coefficients (4 and 9) and variables raised to a power \(x^3\) and \(x\) respectively.

Step 2: Multiply the Coefficients

Multiply the numerical coefficients of both terms: \(4 \times 9 = 36\). This gives us the coefficient for the product of the expressions.

Step 3: Apply Power Rules to Variables

Multiply the variable parts using power rules. When multiplying powers with the same base, you add the exponents: \(x^3 \times x^1 = x^{3+1} = x^4\).

Step 4: Combine the Results

Combine the results from the previous steps. Multiply the new coefficient (36) by the new variable expression \(x^4\): \(36x^4\).

Key Concepts

Power of a Product Rule

When working with algebraic expressions, you may come across the power of a product rule. This rule is crucial for dealing with products within polynomial expressions, especially when variables are involved. The rule states that when you multiply two powers that have the same base, you add the exponents.

For example, in the expression \((x^3) (x^1)\), the base is the same (\(x\)), so you simply add the exponents: \(3 + 1 = 4\). Therefore, the result of \((x^3) (x^1)\) is \(x^4\).

This rule is handy because it simplifies the multiplication of variables, making it easier to manage more complex expressions. It’s a fundamental aspect of algebra that helps to systematically tackle algebraic multiplications.

Polynomial Expression

A polynomial expression is a mathematical phrase that can contain variables, coefficients, and exponents, all combined through addition, subtraction, and multiplication. A polynomial expression could be as simple as \(2x + 3\) or more complex like \(4x^3 + 9x\).

It's important to understand that polynomials are expressions that consist of terms. Each term is made up of a coefficient (a constant number), and a variable raised to a power. For instance, in the term \(4x^3\), \(4\) is the coefficient, and \(x^3\) indicates the variable raised to an exponent.

When performing operations on polynomial expressions, like multiplication, it's key to handle each component correctly. This often involves multiplying coefficients and using exponent rules for variables, such as the power of a product rule.

Variable Coefficient

In algebra, the term variable coefficient refers not to the variable itself, but to the constant number that is multiplied by the variable. In an expression like \(9x\), \(9\) is the coefficient and \(x\) is the variable.

Coefficients determine the number of times a variable is multiplied by itself. Multiplying coefficients simplifies expressions and often involves combining like terms through basic arithmetic.

• For example, when multiplying \(4x^3\) by \(9x\), you first multiply the coefficients: \(4 \times 9 = 36\).
• Then, you use the power of a product rule to multiply the variable parts: \(x^3 \times x = x^4\).

This results in a new term \(36x^4\). Understanding variable coefficients is essential for simplifying expressions and tackling more advanced algebraic concepts effectively.