Problem 25 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using the FOIL method and simplify. $$(3 x+5)(2 x-1)$$

Step-by-Step Solution

Answer

The simplified expression is \(6x^2 + 7x - 5\).

Step 1: Identify the Terms

In the given expression \((3x + 5)(2x - 1)\), the first expression \(3x + 5\) consists of two terms: \(`3x`\) and \(`5`\). The second expression \(2x - 1\) also consists of two terms: \(`2x`\) and \(`-1`\). Prepare to apply the FOIL method to these terms.

Step 2: Multiply the First Terms

Multiply the first terms from each binomial. Start with the first term of each binomial: \(3x\) from \(3x + 5\) and \(2x\) from \(2x - 1\). The product is:\[3x \cdot 2x = 6x^2\].

Step 3: Multiply the Outer Terms

Next, multiply the outer terms. These are the first term of the first binomial and the second term of the second binomial:\(3x\) and \(-1\). The product is:\[3x \cdot (-1) = -3x\].

Step 4: Multiply the Inner Terms

Now, multiply the inner terms. These are the second term of the first binomial and the first term of the second binomial:\(5\) and \(2x\). The product is:\[5 \cdot 2x = 10x\].

Step 5: Multiply the Last Terms

Finally, multiply the last terms in each binomial:\(5\) and \(-1\). The product is:\[5 \cdot (-1) = -5\].

Step 6: Combine the Products

Add all the products from Steps 2 to 5 together to form a single expression:\[6x^2 - 3x + 10x - 5\].

Step 7: Simplify the Expression

Combine the like terms within the expression. The like terms here are the \(x\) terms, \(-3x\) and \(10x\):\[ -3x + 10x = 7x\].So, the final simplified expression is:\[6x^2 + 7x - 5\].

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions form the foundation of algebra. They are combinations of numbers, variables, such as \( x \), \( y \), and arithmetic operations like addition, subtraction, multiplication, and division.
In the expression \((3x + 5)(2x - 1)\), each part inside the parentheses is an algebraic expression.
The expression \(3x + 5\) has the constant term \(5\), and the term \(3x\), which includes a coefficient \(3\) and a variable \(x\).
Similarly, \(2x - 1\) contains a term \(2x\) and a constant \(-1\).

These components are crucial for operations such as addition, subtraction, and multiplication within algebraic contexts. Understanding how these terms interact allows you to apply various algebraic methods, like the FOIL method, more effectively. By mastering the basics of algebraic expressions, you set the stage for solving more complex algebraic problems.

Exploring Binomials

A binomial is a specific type of algebraic expression containing exactly two terms. In our example, \(3x + 5\) and \(2x - 1\) are both binomials.
Binomials are noteworthy because they frequently appear in algebraic problems and solutions. The FOIL method, which stands for First, Outer, Inner, Last, is often used to multiply two binomials.
This method provides a systematic approach to ensure that every term from the first binomial is multiplied by every term from the second binomial.

• The First terms multiply: \(3x\) and \(2x\) make \(6x^2\).
• Outer terms multiply: \(3x\) and \(-1\) make \(-3x\).
• Inner terms multiply: \(5\) and \(2x\) make \(10x\).
• Last terms multiply: \(5\) and \(-1\) make \(-5\).

Understanding binomials and how to manipulate them can greatly simplify complex algebraic expressions through clear patterns and systematic techniques.

Introduction to Polynomials

Polynomials are broader algebraic expressions that can have more than two terms, involving powers of variables and coefficients.
The expression we ended with after simplifying, \(6x^2 + 7x - 5\), is called a polynomial. It's composed of three distinct terms: the quadratic \(6x^2\), the linear \(7x\), and the constant \(-5\).
The degree of this polynomial is determined by the highest power of the variable, which, in this case, is 2 due to \(6x^2\).
Polynomials can be manipulated through various algebraic operations, such as addition, subtraction, and multiplication.

• Identifying the degree helps classify the polynomial.
• Polynomial equations can often be solved or simplified using factoring or other algebraic methods.

In practical applications, polynomials can represent quantities ranging from speed to area, and understanding them is essential for solving a variety of mathematical problems.