Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text {17-28, perform the indicated operations and simplify. } $$ $$ (3 x-4)(x+1) $$

Step-by-Step Solution

Answer

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

Step 1: Set Up the Problem

We need to expand the expression \((3x - 4)(x + 1)\) using the distributive property, also known as the FOIL method for binomials.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \(3x \cdot x = 3x^2\).

Step 3: Multiply the Outer Terms

Multiply the outer terms of the binomials: \(3x \cdot 1 = 3x\).

Step 4: Multiply the Inner Terms

Multiply the inner terms of the binomials: \(-4 \cdot x = -4x\).

Step 5: Multiply the Last Terms

Multiply the last terms of each binomial: \(-4 \cdot 1 = -4\).

Step 6: Combine All Terms

Combine all the terms from the previous steps: \(3x^2 + 3x - 4x - 4\).

Step 7: Simplify the Expression

Combine the like terms \(3x\) and \(-4x\), which gives \(3x^2 - x - 4\). This is the simplified expression.

Key Concepts

Distributive Property

The distributive property is essential for solving multiplication problems involving polynomials or algebraic expressions. It allows you to break down complex expressions into simpler parts that are easier to manage. When you apply the distributive property, you effectively distribute the multiplication of one value across multiple terms within parentheses. In mathematical terms, the property states: \( a(b + c) = ab + ac \). This means you multiply the term outside the parentheses by each term inside the parentheses.
When tackling algebraic expressions, like \( (3x - 4)(x + 1) \), the distributive property helps in multiplying each term in one binomial by every term in the other binomial. This approach makes the process systematic and manageable, reducing potential errors.

Binomials

Understanding binomials is key when dealing with polynomial multiplication. A binomial is an algebraic expression made up of two terms, typically separated by a plus or minus sign.
Examples of binomials include:

• \( (x + 1) \)
• \( (3x - 4) \)

Binomials provide a basis for applying techniques like the FOIL method to expand and simplify more complex expressions. When you multiply binomials, you're essentially applying the distributive property twice, ensuring that each term in the first binomial multiplies every term in the second. This is a foundational step in algebra as it prepares students for dealing with larger polynomials.

FOIL Method

The FOIL method is an efficient approach to multiply two binomials. FOIL stands for First, Outer, Inner, and Last, describing the order in which you multiply the terms. This method is applied to binomials like \( (3x - 4)(x + 1) \), to systematically expand the expression.

• **First**: Multiply the first terms of each binomial, \( 3x \cdot x = 3x^2 \).
• **Outer**: Multiply the outer terms, \( 3x \cdot 1 = 3x \).
• **Inner**: Multiply the inner terms, \( -4 \cdot x = -4x \).
• **Last**: Multiply the last terms, \( -4 \cdot 1 = -4 \).

Once these products are computed, combine them to form a new expression. FOIL is particularly useful because it helps keep track of all products formed during binomial multiplication.

Simplifying Expressions

Simplifying expressions is the process of combining like terms to make an expression easier to understand and use. After applying methods like the distributive property or FOIL, you often end up with terms that can be combined. In algebra, like terms have the same variable raised to the same power.
In the expression \( 3x^2 + 3x - 4x - 4 \), you can combine the like terms \( 3x \) and \(-4x \) to simplify it to \( 3x^2 - x - 4 \). Simplifying is a crucial step in solving algebraic problems as it leads to the final solution that is neat and easy to interpret. This process also aids in understanding the relationships between different parts of the expression, which is especially useful for more advanced algebraic manipulation.