Problem 46 - Algebra 2

Question

Find each product. Classify the result by number of terms. $$ x(2 x)(4 x+1) $$

Step-by-Step Solution

Answer

The product is \(8x^3 + 2x^2\), which is a binomial.

Step 1: Distribute the Monomial

First, distribute the monomial term (in this case, x) to each term inside the parentheses. For the expression \(x(2x)(4x+1)\), we can ignore the \(2x\) for now and focus on multiplying \(x\) by \(4x+1\), which gives us \(4x^2 + x\).

Step 2: Distribute the Remaining Monomial

Next, distribute the remaining monomial \(2x\) to each term from the previous step's result. Multiply \(2x\) by \(4x^2\) to get \(8x^3\), and multiply \(2x\) by \(x\) to get \(2x^2\).

Step 3: Combine the Products

Combine the individual products from Step 2 to obtain the final expression. This will result in: \(8x^3 + 2x^2\).

Step 4: Classify the Result

The final product \(8x^3 + 2x^2\) contains two terms, so it is classified as a binomial.

Key Concepts

Distributive Property

Understanding the distributive property is essential in polynomial multiplication. It's a rule that allows us to multiply a single term by each term inside a set of parentheses. This is exactly what's done when multiplying polynomials together.

For example, when we have an expression such as \(a(b + c)\), the distributive property tells us that the single term outside the parentheses, \(a\), must be multiplied by each term inside: \(ab + ac\). It's an efficient way to simplify expressions before combining like terms. In our textbook problem, we distributed the monomial \(x\) across the binomial \(4x + 1\), resulting in \(4x^2 + x\). This step is crucial to ensure that all parts of the polynomial are multiplied correctly.

Monomial Distribution

When dealing with the term 'monomial distribution', it refers to the process of distributing a single-term polynomial (a monomial) across a multi-term polynomial. A monomial, such as \(2x\), is straightforward since it doesn't involve adding or subtracting within itself.

In our exercise, after using the distributive property to multiply \(x\) with \(4x + 1\), we then distribute the monomial \(2x\) to our new terms \(4x^2\) and \(x\). This step-by-step approach reduces the complexity of the problem, breaking it down into simpler parts that result in \(8x^3 + 2x^2\) after multiplication.

Binomial Terms

Polynomials can have various numbers of terms. When there are two terms, it's called a binomial. Binomials are like the building blocks in many polynomial multiplication tasks because they often come into play when breaking down larger expressions.

Each term in a binomial can be considered individually, making distribution a piece of cake. In our example, the binomial was \(4x + 1\). After applying the distributive property with the monomial \(x\), we multiplied it by each term in the binomial, which paved the way for an easier subsequent distribution with the second monomial \(2x\).

Classifying Polynomials

Classifying polynomials by the number of terms they contain can greatly simplify the understanding of these expressions. Terms are the parts of a polynomial separated by addition or subtraction signs. Polynomials can be monomials (1 term), binomials (2 terms), trinomials (3 terms), or just be referred to as polynomials with more terms.

In the given exercise, after multiplying out all the terms we were left with \(8x^3 + 2x^2\), which has two terms, thus making it a binomial. Recognizing the type of polynomial that results from multiplication can aid in painting a clearer picture of what you're working with and how to proceed with further operations.