Problem 15 - Elementary and Intermediate Algebra

Question

Factor. \(100 m^{2}-1\)

Step-by-Step Solution

Answer

The expression factors to \((10m - 1)(10m + 1)\).

Step 1: Identify the Common Form

The expression is in the form of a difference of squares, which is a common algebraic identity. Recognize that this form is written as \(a^2 - b^2\) and can be factored into \((a-b)(a+b)\).

Step 2: Identify Terms

For the given expression, \(100m^2 - 1\), the term \(100m^2\) can be rewritten as \((10m)^2\) and \(1\) as \(1^2\). This identifies it as a difference of squares with \(a = 10m\) and \(b = 1\).

Step 3: Apply the Difference of Squares Formula

Substitute \(a = 10m\) and \(b = 1\) into the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). This leads to \((10m - 1)(10m + 1)\).

Step 4: Write the Factored Form

After applying the difference of squares formula, the factored form of the expression \(100m^2 - 1\) is \((10m - 1)(10m + 1)\).

Key Concepts

Factoring Expressions

Factoring expressions is an essential skill in algebra that involves rewriting an equation as a product of its factors. It simplifies complex polynomial expressions and makes solving equations easier.
In the context of the original exercise, we have an expression that needs to be factored:

• Recognize the type of expression: In our case, it is a quadratic expression, specifically a difference of squares.
• Identify components: Look for patterns or structures that match known formulas, allowing you to restructure the expression as a multiplication of simpler terms.
• Use formulas: Applying formulas like the difference of squares helps in factoring effectively.

Factoring is essential because it can reveal the roots of an equation, simplify computations, and make seemingly difficult algebra problems more manageable.

Algebraic Identities

Algebraic identities are known algebraic expressions that are universally true and can be used to simplify and solve expressions. Understanding and using these identities is key to mastering algebra.
The difference of squares formula is an important algebraic identity:

• It states that for two numbers or expressions, say \(a\) and \(b\), \(a^2 - b^2 = (a - b)(a + b)\).
• In the original exercise, this identity was leveraged to factor the expression \(100m^2 - 1\) by identifying \(a = 10m\) and \(b = 1\).
• Using this identity simplifies complex expressions rapidly and efficiently.

Mastering algebraic identities allows students to quickly recognize patterns and simplifies the process of equation manipulation.

Polynomials

Polynomials are expressions made up of variables and coefficients, consisting of terms that can be added, subtracted, multiplied, and subjected to exponents. Understanding polynomials is crucial as they form the foundation for many algebraic structures.
Here’s what makes them special:

• Degree of polynomials: This indicates the highest power of the variable in the expression.
• Types: Include monomials, binomials, trinomials, and so on, classified based on the number of terms.
• Operations: Polynomials can be added, subtracted, and multiplied, and they can also be factored, which is critical for simplifying expressions.

The expression \(100m^2 - 1\) is a binomial because it has two terms, and is further characterized as a quadratic polynomial because the highest exponent on the variable \(m\) is two. Understanding these classifications assists in identifying appropriate factoring techniques, such as using algebraic identities.