Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((x+5)(x-5)\)

Step-by-Step Solution

Answer

\(x^2 - 25\)

Step 1: Apply FOIL method

The FOIL method stands for First, Outside, Inside, Last. It's a way for remembering how to expand the product of two binomials. So apply it to \((x+5)(x-5)\). First, multiply the first terms in each binomial together.

Step 2: Multiply the outer terms

Next, multiply the outer terms together. The outer terms are \(x\) and \(-5\). Their product is \(-5x\).

Step 3: Multiply the inner terms

Now, multiply the inner terms together, which are \(5\) and \(x\). Their product is \(5x\).

Step 4: Multiply the last terms

Finally, multiply the last terms of both binomials together. The last terms are \(5\) and \(-5\). Their product is \(-25\).

Step 5: Combine like terms

Combine like terms. The like terms here are \(-5x\) and \(5x\). So, the combined terms cancel out each other and the answer is \(x^2 - 25\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. The degree of a polynomial is determined by the highest exponent of its variable. For example, in the expression \(x^2 - 25\), \(x^2\) is the highest power, making it a second-degree polynomial. Polynomials can have multiple terms:

• Monomials: Single term, like \(3x\).
• Binomials: Two terms, like \(x+5\).
• Trinomials: Three terms, like \(x^2 + 2x + 1\).

Each term in a polynomial can also contain coefficients, which are numbers multiplying the variable part. In the polynomial \(x^2 - 25\), the coefficient of \(x^2\) is 1, and the constant \(-25\) is also a term but without any variable. Understanding polynomials is vital because they appear in various areas of mathematics and real-world applications, such as physics and economics.

Binomials

Binomials are a specific type of polynomial consisting of exactly two terms. They are often expressed in the form \(a + b\), where \(a\) and \(b\) can be numbers or expressions with variables. A common operation involving binomials is multiplying them, which can be efficiently done using the FOIL method. In the expression \((x+5)(x-5)\), we see a classic example of two binomials.
The FOIL method helps by systemizing the multiplication process:

• First: Multiply the first terms: \(x \times x = x^2\).
• Outside: Multiply the outer terms: \(x \times -5 = -5x\).
• Inside: Multiply the inner terms: \(5 \times x = 5x\).
• Last: Multiply the last terms: \(5 \times -5 = -25\).

Notice how the middle terms \(-5x\) and \(5x\) cancel each other out. This simplifies the expression \((x+5)(x-5)\) to \(x^2 - 25\), showcasing the difference of squares identity.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). They form the basic building blocks of algebra. Unlike equations, which show equality between two expressions, algebraic expressions do not include an equality sign.
An important application of algebraic expressions is in forming and solving polynomials. Expressions can further classify into monomials, binomials, trinomials, and polynomials based on the number of terms. The exercise \((x+5)(x-5)\) involves multiplying two algebraic expressions (binomials), which results in the expression \(x^2 - 25\).
When working with algebraic expressions, simplifying them by combining like terms or applying identities, such as the difference of squares, can make calculations more straightforward. Mastery of algebraic expressions allows for solving complex mathematical problems and understanding the relationships between quantities.