Problem 101 - Intermediate Algebra

Question

Perform each indicated operation. See Sections 1.4 and 5.4 $$ (2 x+3)(x-5) $$

Step-by-Step Solution

Answer

The result of \((2x + 3)(x - 5)\) is \(2x^2 - 7x - 15\).

Step 1: Identify the Operation

We need to perform the multiplication of two binomials: \((2x + 3)(x - 5)\). To solve this, we'll apply the distributive property (also known as the FOIL method for binomials), which stands for First, Outer, Inner, and Last terms.

Step 2: Multiply First Terms

The first terms in each binomial are \(2x\) and \(x\). Multiply these:\(2x \cdot x = 2x^2\).

Step 3: Multiply Outer Terms

The outer terms are \(2x\) and \(-5\). Multiply these:\(2x \cdot (-5) = -10x\).

Step 4: Multiply Inner Terms

The inner terms are \(3\) and \(x\). Multiply these: \(3 \cdot x = 3x\).

Step 5: Multiply Last Terms

The last terms in each binomial are \(3\) and \(-5\). Multiply these: \(3 \cdot (-5) = -15\).

Step 6: Combine All Terms

Now add all the results from the previous steps together:\(2x^2 - 10x + 3x - 15\).

Step 7: Simplify the Expression

Combine the like terms (terms with \(x\)):\(-10x + 3x = -7x\).So the simplified expression is:\(2x^2 - 7x - 15\).

Key Concepts

FOIL Method

The FOIL Method is a popular technique used to multiply two binomials quickly and efficiently. It specifically applies to expressions of the form \((a+b)(c+d)\) and helps ensure that each component of the binomials is considered in the calculation.

**What Does FOIL Stand For?**

• First: Multiply the first terms of each binomial, which, in the example \((2x+3)(x-5)\), are \(2x\) and \(x\). The result is \(2x^2\).
• Outer: Multiply the outer terms, means multiplying the first term of the first binomial with the last term of the second binomial, so \(2x\) and \(-5\) give \(-10x\).
• Inner: Multiply the inner terms, which is \(3\) and \(x\), resulting in \(3x\).
• Last: Finally, multiply the last terms of each binomial, \(3\) and \(-5\), yielding \(-15\).

After applying FOIL, the results are gathered and combined: \(2x^2 - 10x + 3x - 15\), and simplified to \(2x^2 - 7x - 15\).

Remember, while FOIL is particularly useful for binomials, understanding the underlying principles will help with more complex expressions as well.

Distributive Property

The Distributive Property is a fundamental cornerstone in algebra that allows for the multiplication of a single term by each term within a parenthesized expression. When you multiply each term of a binomial by another binomial, the process might look complex initially, but it breaks down into manageable steps.

**Understanding the Distributive Property**

• It involves moving systematically across each term of one binomial and multiplying it by every term in the second binomial.
• In the problem \((2x+3)(x-5)\), the property is applied by multiplying \(2x\) with both \(x\) and \(-5\), and then \(3\) with \(x\) and \(-5\).
• The products \(2x \cdot x = 2x^2\), \(2x \cdot (-5) = -10x\), \(3 \cdot x = 3x\), and \(3 \cdot (-5) = -15\) are derived by following this method.

By collecting all these results and combining them into one expression, we see the distributive property in action: \(2x^2 - 10x + 3x - 15\).

This process illustrates how the distributive property applies not just to simple arithmetic but extends into more complex algebraic contexts.

Algebraic Expressions

Algebraic expressions are essential building blocks of algebra that allow us to explore relationships between numbers using variables. They consist of terms, which are a combination of constants (fixed numbers) and variables (symbols that represent numbers) connected by operators like addition and subtraction.

**Components of an Algebraic Expression**

• **Terms:** These can be constants, variables, or coefficients multiplied by a variable, like \(2x\) or \(-5\).
• **Operators:** These include addition \(+\) and subtraction \(-\), which combine terms.

In our earlier multiplication \((2x+3)(x-5)\), we work with the expression terms \(2x^2\), \(-10x\), \(3x\), and \(-15\). The final expression\(2x^2 - 7x - 15\)reflects a simplified form achieved by combining like terms \(-10x + 3x\). This step is crucial in ensuring clarity and simplicity in algebraic structures.

Respecting these components leads to an enhanced understanding of how to manipulate and solve expressions in algebra, simplifying the transition to solving equations and more complex algebraic structures.