Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated operations and simplify. \((3 x-4)(x+1)\)

Step-by-Step Solution

Answer

The simplified expression is \(3x^2 - x - 4\).

Step 1: Expand the Binomials

To expand the two binomials \((3x - 4)(x + 1)\), we use the distributive property or the FOIL method. We multiply each term in the first binomial by each term in the second binomial:- First: \((3x) \cdot (x) = 3x^2\)- Outer: \((3x) \cdot (1) = 3x\)- Inner: \((-4) \cdot (x) = -4x\)- Last: \((-4) \cdot (1) = -4\)

Step 2: Combine Like Terms

Now that we have expanded the expression, we combine the like terms:The expression is \(3x^2 + 3x - 4x - 4\).Combine the \(3x\) and \(-4x\) terms:\(3x^2 + 3x - 4x - 4 = 3x^2 - x - 4\).The simplified expression is \(3x^2 - x - 4\).

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle that allows us to simplify expressions and solve equations. When dealing with expressions like \[(3x-4)(x+1)\],this property is applied to multiply each term inside one binomial by each term in the other binomial.To break it down, consider how this principle works:

• Multiply every term in the first group by each term in the second group.
• This ensures that each term is accounted for and helps in transforming complex expressions into more manageable components.

In our example, you start by multiplying 3x by both x and 1, followed by multiplying -4 by both x and 1. The goal is to make sure that no term is left out. As a result, you achieve a fully expanded form that you can later simplify. Remembering the distributive property is key when expanding binomials and transforming polynomials.

FOIL Method

The FOIL method is a specialized technique used to expand binomials, particularly when you have a simple expression like \[(3x-4)(x+1)\].FOIL stands for:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

Using this approach on the given expression, we perform the following operations:

• First: \( (3x) \cdot (x) = 3x^2 \)
• Outer: \( (3x) \cdot (1) = 3x \)
• Inner: \( (-4) \cdot (x) = -4x \)
• Last: \( (-4) \cdot (1) = -4 \)

By the end of this step, you will have an expanded polynomial. The FOIL method is efficient for straightforward binomial multiplications and can be a useful mnemonic to keep this process clear.

Polynomial Simplification

Polynomial simplification is the process of transforming a polynomial into its simplest form, making it easier to understand and work with in further calculations.After expanding a polynomial using methods like the distributive property or FOIL, you'll likely end up with an expression that needs cleaning up. For example, once the binomials \[(3x-4)(x+1)\] are expanded, you are left with the expression \[3x^2 + 3x - 4x - 4\]. To simplify:

• Combine like terms: Look for terms with the same variables raised to the same power—they can be added or subtracted as needed.
• In our case, the terms \(3x\) and \(-4x\) are like terms, which combine to form \(-x\).

This gives us the final simplified polynomial: \[3x^2 - x - 4\].Simplifying polynomials is an important skill, as it allows you to work with more efficient and manageable expressions for further computations and analyses.