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Chapter 1: Problem 17

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

Short Answer

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

Step by step solution

01

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: \((-4) \cdot (1) = -4\)
02

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 \(3x^2 - x - 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攖hey 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.

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