Problem 56 - 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. $$(x+6)\left(2 x^{2}-x-7\right)$$

Step-by-Step Solution

Answer

The product is \(2x^3 + 11x^2 - 13x - 42\).

Step 1: Understanding the Problem

We need to find the product of the binomial \((x+6)\) and the trinomial \((2x^2 - x - 7)\). To solve this, we will apply the distributive property to multiply each term in the binomial by each term in the trinomial.

Step 2: Distribute the First Term of the Binomial

Start by distributing the first term in the binomial, \(x\), to each term in the trinomial \((2x^2 - x - 7)\). This results in: \(x \cdot 2x^2 = 2x^3\), \(x \cdot (-x) = -x^2\), and \(x \cdot (-7) = -7x\).

Step 3: Distribute the Second Term of the Binomial

Next, distribute the second term in the binomial, \(6\), to each term in the trinomial \((2x^2 - x - 7)\). This results in: \(6 \cdot 2x^2 = 12x^2\), \(6 \cdot (-x) = -6x\), and \(6 \cdot (-7) = -42\).

Step 4: Combine Like Terms

Now, combine all the terms obtained from the distributions: \(2x^3 - x^2 - 7x\) (from the first term, \(x\)) and \(12x^2 - 6x - 42\) (from the second term, \(6\)). Combining like terms gives: \(2x^3 + (12x^2 - x^2) + (-7x - 6x) - 42\).

Step 5: Simplify the Expression

Simplify the combined expression: - Combine the \(x^2\) terms: \(12x^2 - x^2 = 11x^2\)- Combine the \(x\) terms: \(-7x - 6x = -13x\)- The constant term remains at \(-42\).The final simplified expression is: \(2x^3 + 11x^2 - 13x - 42\).

Key Concepts

Understanding a Binomial

A binomial is a polynomial with exactly two terms. Each term in a binomial is separated by a plus sign (+) or a minus sign (-). For example, in the exercise, we're dealing with the binomial \(x + 6\). This means there are two parts:

• \(x\)
• \(6\)

Each term can stand alone or be multiplied by other terms. When multiplying binomials by other polynomials, each term in the binomial must be multiplied by every term in the other polynomial. Understanding how to manage these terms is crucial for polynomial operations.

Exploring the Trinomial

A trinomial is a type of polynomial that has three distinct terms. These terms are typically separated by plus or minus signs as well. In the problem, we have the trinomial \(2x^2 - x - 7\). Its three terms are:

• \(2x^2\)
• \(-x\)
• \(-7\)

Each term plays a key role when you're performing polynomial operations. When multiplying a trinomial by another polynomial, such as a binomial, it's vital to apply each multiplication carefully and methodically to cover all possible combinations of terms.

The Distributive Property Simplified

The distributive property is a fundamental principle used in algebra to simplify expressions. It says you must multiply each term inside a parenthesis by a term outside the parenthesis. For instance, in this exercise, when multiplying \((x + 6)\) by \(2x^2 - x - 7\), each term in \(x + 6\) becomes a separate multiplier:

• Multiply \(x\) by every term in the trinomial.
• Multiply \(6\) by each term in the trinomial.

By following the distributive property step by step, you ensure that no term is left out in the process, which allows for the complete expansion and combination of terms.

Simplifying with Like Terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power. In simplifying the product of a binomial and a trinomial, such as in this exercise, recognizing and combining like terms is essential. After performing all the necessary distributions, you'll encounter terms like:

• \(x^3\) terms
• \(x^2\) terms
• \(x\) terms
• Constant terms

To simplify, combine coefficients of these like terms:

• Combine \(x^2\) terms: \(12x^2 - x^2 = 11x^2\).
• Combine \(x\) terms:\(-7x - 6x = -13x\).

This results in cleaner and more manageable expressions, making it easier to solve arithmetic problems.