Problem 25 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 4 a^{2}-25 $$

Step-by-Step Solution

Answer

Answer: The factors of the expression \(4a^2 - 25\) are \((2a + 5)\) and \((2a - 5)\).

Step 1: Identify the Difference of Squares

Recognize that given expression is a difference of squares: $$4a^2 - 25 = (2a)^2 - (5)^2$$

Step 2: Apply the Difference of Squares Formula

Use the general formula for the difference of squares to factor the expression: $$A^2 - B^2 = (A + B)(A - B)$$ In this case, \(A = 2a\) and \(B = 5\). So, the factored expression will be: $$(2a + 5)(2a - 5)$$

Key Concepts

Difference of Squares

Understanding the difference of squares is crucial when it comes to factoring certain types of algebraic expressions. This mathematical phenomenon occurs when a binomial is composed of two perfect squares separated by a subtraction sign. The formula to factor the difference of squares is \(A^2 - B^2 = (A + B)(A - B)\), where \(A\) and \(B\) are any expressions for which \(A^2\) and \(B^2\) are defined.

Now, why is this valuable? Because it provides a quick and systematic way to simplify expressions that can otherwise seem complex. In our example, the expression \(4a^2 - 25\) is recognized as the difference of squares because \(4a^2\) is a perfect square (\((2a)^2\)) and \(25\) is also a perfect square (\((5)^2\)). The subtraction sign between them is your signal to apply this special factoring technique.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols that represent a certain quantity. In the context of polynomial factoring, we focus on expressions that can be rewritten into a product of polynomials. The expression \(4a^2 - 25\) is a simple algebraic expression where \(a\) is the variable and \(4\) and \(25\) are constant coefficients.

It's important to recognize different forms of algebraic expressions, as this aids in identifying the most effective factoring technique. For example, realizing that \(4a^2 - 25\) is a difference of squares — not just an arbitrary subtraction of terms — provides a clear path to simplification. Recognizing these forms is like discerning patterns in a puzzle, making the complex world of algebra more approachable.

Polynomial Factoring

Polynomial factoring is a method used to express a polynomial as the product of two or more simpler polynomials. The factoring process simplifies the polynomial and can make other operations, such as solving equations, much easier. Factoring is like breaking down a complex structure into its essential building blocks.

There are several factoring techniques, but for a difference of squares, as in our exercise \(4a^2 - 25\), the process is quick and intuitive. After recognizing the expression as a difference of squares, we use the formula \(A^2 - B^2 = (A + B)(A - B)\) to write the factored form. Each factor represents a potential 'building block' of the original expression. For more complex polynomials, factoring might involve finding the greatest common factor (GCF), grouping, or utilizing special products such as the sum of cubes, but each serves the same purpose: to simplify and reveal structure within algebraic expressions.