Problem 58 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(3 x+4)\left(2 x^{2}-2 x-6\right)$$

Step-by-Step Solution

Answer

The product is \(6x^3 + 2x^2 - 26x - 24\).

Step 1: Identify the expression structure

The expression to be multiplied is a combination of a binomial, \(3x + 4\), and a trinomial, \(2x^2 - 2x - 6\). Our task is to find the product of these two polynomials.

Step 2: Apply the distributive property

Use the distributive property to multiply each term in the binomial with every term in the trinomial. That means multiplying \(3x\) by each term in the trinomial and then \(4\) by each term in the trinomial.

Step 3: Multiply the first term (3x) with each trinomial term

Calculate the following products:\(3x \cdot 2x^2 = 6x^3\), \(3x \cdot (-2x) = -6x^2\), and \(3x \cdot (-6) = -18x\).

Step 4: Multiply the second term (4) with each trinomial term

Calculate the following products: \(4 \cdot 2x^2 = 8x^2\), \(4 \cdot (-2x) = -8x\), and \(4 \cdot (-6) = -24\).

Step 5: Combine and simplify all terms

Now, add all the products from steps 3 and 4 together: \(6x^3 - 6x^2 - 18x + 8x^2 - 8x - 24\). Combine like terms: \[6x^3 + (-6x^2 + 8x^2) + (-18x - 8x) - 24 = 6x^3 + 2x^2 - 26x - 24.\]

Key Concepts

Binomials

A binomial is a type of polynomial that consists of exactly two terms. These terms are typically joined by a plus or minus sign. In the context of polynomial algebra, understanding binomials is crucial because they serve as the building blocks for more complex expressions.

Consider the binomial expression in our problem: \(3x + 4\). Here, **3x** and **4** are the two terms that form the binomial. Each term is a simple polynomial on its own, with **3x** having a degree of 1 (because the variable x is raised to the power of 1) and **4** being a constant term.

When multiplying a binomial by another polynomial, we'll use specific techniques such as the distributive property to break down and simplify the expression. Recognizing this structure helps streamline the multiplication process.

Trinomials

A trinomial is a polynomial that contains three distinct terms. Like binomials, trinomials come together by connecting terms with plus or minus signs. In our exercise, the trinomial used is \(2x^2 - 2x - 6\). Understanding its components helps simplify the process of multiplication.

In the trinomial \(2x^2 - 2x - 6\):

• **2x^2** is the term of highest degree (degree 2)
• **-2x** is a linear term (degree 1)
• **-6** is a constant term (degree 0)

Breaking down trinomials and identifying each component is essential. This knowledge allows us to systematically use the distributive property, and later, to effectively combine and simplify results. This understanding ensures accurate polynomial multiplication, avoiding common pitfalls in algebra.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It's especially useful in multiplying polynomials like binomials and trinomials, which involves distributing each term of the first polynomial over every term of the second.

In our exercise, we apply the distributive property by multiplying each term in the binomial \(3x + 4\) with each term in the trinomial \(2x^2 - 2x - 6\).

This step is carried out in two parts:

• First, multiply \(3x\) by each term in the trinomial: \(3x \cdot 2x^2, 3x \cdot (-2x), 3x \cdot (-6)\).
• Next, multiply \(4\) by each term in the trinomial: \(4 \cdot 2x^2, 4 \cdot (-2x), 4 \cdot (-6)\).

By applying the distributive property, we break down complex multiplications into manageable parts, ensuring we capture every interaction between terms.

Polynomial Multiplication Steps

The multiplication of polynomials is a step-by-step process where each term of one polynomial is multiplied with every term of the other. It is an expansion process, ensuring that all products are accounted for. Following organized steps makes this task more approachable.

1. **Identify the Polynomials**: Determine the structures we're multiplying, such as a binomial and a trinomial in this case.
2. **Apply the Distributive Property**: For every term in the first polynomial, multiply it by each term in the second polynomial. This expands the expression into individual products.
3. **Calculate the Individual Products**: Compute each product separately. For example, in our exercise:

• Multiply \(3x\) by each term of the trinomial
• Multiply \(4\) by each term of the trinomial

4. **Combine Like Terms**: This involves adding together terms that have identical variable components. In our example: \(-6x^2\) and \(8x^2\) are like terms, combined to form \(2x^2\).

Following these systematic steps ensures that polynomial multiplication is consistent and accurate, leading to a simplified result of \(6x^3 + 2x^2 - 26x - 24\). This methodical approach eliminates errors and facilitates understanding.