Problem 69 - Elementary and Intermediate Algebra

Question

Multiply. $$ \left(2 x^{2}\right)(5 x) $$

Step-by-Step Solution

Answer

The product is \(10x^3\).

Step 1: Identify Components

First, identify the components in the expression \((2x^2)(5x)\). We have a monomial \(2x^2\) and another monomial \(5x\).

Step 2: Multiply Coefficients

Multiply the coefficients of the monomials. The coefficient of the first monomial is 2, and the second is 5. So, multiply 2 and 5 to get 10.

Step 3: Multiply the Variables

Next, multiply the variable parts. The first monomial has \(x^2\) and the second monomial has \(x^1\). Multiply these to get \(x^{2+1} = x^3\).

Step 4: Combine Results

Combine the results from the previous steps to obtain the final result. We have the coefficient from step 2 as 10, and the variable from step 3 as \(x^3\), so the final result is \(10x^3\).

Key Concepts

Monomials

A monomial is a basic building block in algebra. It is an expression that consists of a single term, and importantly, it can be a number, a variable, or a product of numbers and variables. A monomial does not have addition or subtraction involved. For example, in the given exercise, both \(2x^2\) and \(5x\) are monomials.Monomials are used to form more complex algebraic expressions. In our context:

• \(2x^2\) consists of a coefficient \(2\) and a variable part \(x^2\).
• \(5x\) consists of a coefficient \(5\) and a variable part \(x\).

One of the operations you can perform with monomials is multiplication, as seen in the exercise. Multiplying monomials involves multiplying their coefficients separately from their variable parts.

Coefficients

In algebra, a coefficient is a number placed in front of a variable in a monomial and represents the factor that multiplies that variable. Coefficients provide a way to scale the variables.For example:

• In the monomial \(2x^2\), the coefficient is 2.
• In the monomial \(5x\), the coefficient is 5.

To multiply monomials, you first multiply the coefficients of each monomial together. In our exercise, you calculate the product of \(2\) and \(5\) to obtain \(10\). This product is the coefficient in the resulting monomial after multiplication. Thus, the result involves scaling up by the factor \(10\). Coefficients are crucial when comparing sizes of terms or creating balanced equations.

Variables

Variables are symbols used to represent numbers in algebraic expressions. They are typically depicted as letters, which can stand for any number or represent fixed numbers as constraints arise. In our exercise, both monomials share the variable \(x\).In multiplication of monomials:

• Consider their powers. Power represents how many times the variable is used as a factor.
• In the monomial \(2x^2\), \(x^2\) means \(x\) is used twice.
• In \(5x\), it is the same as \(x^1\), meaning it appears once.

To multiply variable parts, add their exponents. So \(x^2 \times x^1 = x^{2+1} = x^3\). This addition of exponents springs from the rule of multiplying powers with the same base in algebra. Exponents indicate the variable's magnitude in a term or expression, and manipulating them is a significant skill in algebra.