Problem 16 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ a^{2}-25 $$

Step-by-Step Solution

Answer

Answer: \((a + 5)(a - 5)\)

Step 1: Identify the difference of squares

We are given the expression \(a^2 - 25\). Notice that this is a difference of squares, since both terms are perfect squares. We have \(a^2\) as the square of \(a\) and \(25\) as the square of \(5\).

Step 2: Factor the expression

Knowing it is a difference of squares, we can now factor the expression using the formula \(A^2 - B^2 = (A + B)(A - B)\). In our case, \(A = a\) and \(B = 5\).

Step 3: Write the final answer

Applying the formula, we get \((a + 5)(a - 5)\), which is the factored form of \(a^2 - 25\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. A variable is a symbol, usually a letter, that represents an unknown value. For example, in the expression \(a^{2}-25\), \(a\) is the variable, and when combined with numbers and operations, it forms an algebraic expression. Understanding algebraic expressions is the foundation of algebra and plays a pivotal role in solving equations and factoring polynomials.

Recognizing the structure of algebraic expressions helps in applying the appropriate methods to simplify or manipulate them. In the expression \(a^{2}-25\), the structure is that of a difference of two terms, which signifies subtraction. Seeing this structure is essential as it guides us towards specific factoring techniques such as factoring a difference of squares.

Perfect Squares

Perfect squares are numbers or expressions that are the squares of integers or algebraic terms. For instance, \(25\) is a perfect square as it is \(5^{2}\), the square of \(5\). Similarly, in algebra, \(a^{2}\) is a perfect square because it represents \(a\) times \(a\).

Being able to identify perfect squares within algebraic expressions allows us to apply simplification techniques and factoring strategies effectively. For the exercise \(a^{2}-25\), noticing that \(a^{2}\) and \(25\) are both perfect squares directly leads to the method of factoring by recognizing the pattern as a difference of squares.

Binomials

Binomials are algebraic expressions with exactly two terms, for example, \(a^2\) and \(25\) in the expression \(a^{2}-25\). In mathematics, it's common to work with binomials in various operations, including addition, subtraction, and particularly, factoring. When binomials show a specific pattern, like the difference of squares, they can be factored in a more streamlined manner.

Understanding the properties of binomials is crucial for factoring them efficiently. Recognizing that a binomial could represent a difference of squares, as in the example above, simplifies the process of breaking it down into its factorized form, \(a + 5)(a - 5)\).

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or factors. Factors of polynomials are algebraic expressions that, when multiplied together, give back the original polynomial. As seen in the exercise \(a^{2}-25\), factoring involves recognizing patterns that match known formulas or identities, such as the difference of squares formula \(A^{2} - B^{2} = (A + B)(A - B)\).

Mastering factoring techniques is fundamental in solving algebraic equations, simplifying expressions, and finding polynomial roots. By identifying the structure of the polynomial as a difference of squares, we can easily factor it into \(a + 5)(a - 5)\), unveiling a more digestible pair of binomials useful for further calculations or insights into the properties of the original polynomial.