Problem 38 - Beginning and Intermediate Algebra

Question

Factor completely. $$d^{2}-81$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is: \(d^2 - 81 = (d+9)(d-9)\).

Step 1: Identify a and b

In our expression, \(d^2 - 81\), the a and b values are: a = d (since \(a^2 = d^2\)) b = 9 (since \(b^2 = 81\))

Step 2: Apply the Difference of Squares Formula

Now we apply the difference of squares formula: \(a^2 - b^2 = (a+b)(a-b)\) So, for our given expression, we have: \(d^2 - 81 = (d+9)(d-9)\)

Step 3: Final Answer

The completely factored form of the given expression is: \(d^2 - 81 = (d+9)(d-9)\)

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In algebra, these symbols (often represented as letters) stand for numbers whose exact values may not always be known. The beauty of algebra is in its ability to convey general relationships and solve for unknowns. In our exercise about factoring polynomials, we use algebra to transform the expression into a simpler form. This is achieved by applying certain rules and operations. The purpose of factoring in algebra, particularly polynomial expressions, is to simplify them or find solutions to equations. Understanding the rules of algebra is crucial, as it forms the foundation for tackling more complex equations and expressions. By mastering these basic principles, students can easily transition into more advanced mathematical challenges later on.

Exploring the Difference of Squares

The difference of squares is a unique algebraic concept used to simplify polynomial expressions, specifically when subtracting one square from another. It follows a simple formula, which states: \[a^2 - b^2 = (a+b)(a-b)\] This key formula tells us that the difference between two squared terms can be expressed as two binomials multiplied together. In the exercise, the expression \(d^2 - 81\) is a classic example of the difference of squares; here, \(d^2\) is the square of \(d\), and \(81\) is the square of \(9\).

• Identify the squared terms. In this case, \(a = d\) and \(b = 9\).
• Apply the formula to transform the expression into \((d + 9)(d - 9)\).

Using the difference of squares technique allows for simpler expressions and can be a powerful tool in solving various algebraic problems.

Working with Polynomial Expressions

Polynomial expressions are fundamental elements in algebra. A polynomial is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. For instance, \(d^2 - 81\) is a simple polynomial expression involving the variable \(d\) raised to a power.When tackling polynomial expressions, knowing how to factor them is crucial. Factoring polynomials involves breaking down a complicated expression into simpler terms, or factors, that, when multiplied together, yield the original expression. Here are the key points of factoring polynomial expressions:

• Identify the type of polynomial. For example, a difference of squares or a quadratic.
• Use appropriate factoring techniques like the difference of squares for specific forms.
• Simplify the expression where possible.

Understanding polynomial expressions and being able to factor them effectively can significantly aid in simplifying problems and finding solutions to algebraic equations.