Problem 143 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(25 x^{2}-4 a^{2}\)

Step-by-Step Solution

Answer

The expression factors to \((5x - 2a)(5x + 2a)\).

Step 1: Recognize the Structure

Recognize that the given expression \(25x^{2} - 4a^{2}\) is a difference of squares. A difference of squares takes the form \(A^{2} - B^{2} = (A - B)(A + B)\).

Step 2: Identify the Squares

Identify the terms \(A^{2}\) and \(B^{2}\). Here, \(A^{2} = (5x)^{2}\) and \(B^{2} = (2a)^{2}\). Thus, \(A = 5x\) and \(B = 2a\).

Step 3: Apply the Difference of Squares Formula

Using the identified squares, apply the difference of squares formula to factor the expression: \(25x^{2} - 4a^{2} = (5x - 2a)(5x + 2a)\).

Key Concepts

Difference of Squares

The difference of squares is a specific technique used when factoring particular types of polynomial expressions. It comes from recognizing the special binomial pattern of the form \(A^2 - B^2\). This pattern applies because it can be rewritten using the identity: \((A - B)(A + B)\). This means that when you have a subtraction between two squared terms, it can be neatly factored into a multiplication of two different binomials. You accomplish this by taking the square root of each squared term.
For instance, if you encounter an expression like \(25x^2 - 4a^2\), you should notice it fits the difference of squares pattern where \(25x^2\) is \((5x)^2\) and \(4a^2\) is \((2a)^2\). Hence, using the formula:

• Factor it into two expressions: \((5x - 2a)\) and \((5x + 2a)\).
• Each of these factors is a binomial with the terms squared within the original equation.

This approach showcases the power of identifying patterns in algebra which provides an efficient way to solve seemingly complex problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. Understanding these expressions involves knowing how they can be simplified and manipulated through various techniques, like factoring and expanding. Algebraic expressions serve as building blocks in solving equations and modeling real-world situations.
They consist of:

• Terms: which can be numbers, variables, or a constant multiplied by variables like \(5x^2\) or \(4a^2\).
• Operators: which determine the action between terms, such as addition \((+)\), subtraction \((-\)))\, multiplication \((\cdot)\), and division \((/)\).

In our example, when we have the expression \(25x^2 - 4a^2\), it is vital to recognize its format as an algebraic expression. From there, you can apply factoring techniques to simplify or solve it, turning a potentially complicated expression into a more straightforward multiplication problem by utilizing the difference of squares.

Polynomial Equations

Polynomial equations are equations that involve a polynomial, which is an expression of more than two algebraic terms, especially the sum of several terms that include different powers of the same variable(s). The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\). The degree of the polynomial is dictated by the highest power of the variable.
Solving polynomial equations often involves factoring the expression where possible. This makes working out the roots or solutions more straightforward.

• For instance, factoring \(25x^2 - 4a^2\) simplifies the process of solving as it becomes \((5x - 2a)(5x + 2a) = 0\).
• By setting each factor equal to zero, you can solve for the variable \(x\).

Factoring plays a crucial role in mathematics as it breaks down complex polynomial equations into products of simpler binomial expressions, facilitating easier computations or graphing of solutions.