Problem 36 - Algebra A Combined Function

Question

Factor each completely. $$ 25 x y^{2}-4 x $$

Step-by-Step Solution

Answer

The factored form is \(x(5y - 2)(5y + 2)\).

Step 1: Identify Common Factors

First, look for any common factors in both terms of the expression. The expression given is \(25xy^2 - 4x\). Both terms contain the factor \(x\). Factor this out to get: \(x(25y^2 - 4)\).

Step 2: Recognize as a Difference of Squares

Notice that the expression inside the parentheses, \(25y^2 - 4\), is a difference of squares. A difference of squares can be factored using the formula \(a^2 - b^2 = (a-b)(a+b)\). Here, identify \(a^2 = (5y)^2\) and \(b^2 = 2^2\).

Step 3: Apply Difference of Squares Formula

Substitute \(a = 5y\) and \(b = 2\) into the difference of squares formula: \((5y)^2 - (2)^2 = (5y - 2)(5y + 2)\). Thus, the expression becomes: \(x(5y - 2)(5y + 2)\).

Key Concepts

Common Factors

In algebra, identifying the common factor is a crucial initial step in factoring polynomials. A common factor refers to a number or variable that is shared by each term in an expression. To spot it, examine each term for shared components that can be factored out. For example, when given an expression like \(25xy^2 - 4x\), you want to identify which variables or numbers are present in both terms. Both terms contain the variable \(x\). Therefore, factoring out \(x\) leaves us with \(x(25y^2 - 4)\). This process simplifies expressions, making the next steps in factoring more manageable. Finding common factors can involve:

• Looking for shared numbers: such as \(2, 3, 5\), etc., in each term.
• Identifying shared variables: such as \(x, y\), etc., present in every term.

Recognizing these factors provides a foundation for further simplification of the polynomial.

Difference of Squares

Once common factors have been extracted from a polynomial, check if what's left can be represented as a difference of squares. This specific form involves two square terms with subtraction in between. The expression \(a^2 - b^2\) is a difference of squares. It can be factored using the formula: \(a^2 - b^2 = (a-b)(a+b)\). In the expression \(25y^2 - 4\), we can identify it as a difference of squares because:

• The first term \(25y^2\) is a perfect square, which is \((5y)^2\).
• The second term \(4\) is also a perfect square, namely \((2)^2\).

Thus, \(25y^2 - 4\) is equivalent to \((5y - 2)(5y + 2)\). Recognizing and applying the difference of squares makes the polynomial simpler to solve or evaluate.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (like addition, subtraction, multiplication, and division). These expressions can represent various mathematical situations and need simplifying or factoring for easy use. Factoring transforms intricate expressions into a product of simpler factors. Considerations while factoring include:

• Checking for common factors first, simplifying the expression.
• Recognizing patterns like the difference of squares to further break down the expression.

For example, with the expression \(25xy^2 - 4x\), the process starts with extracting the common factor \(x\), then recognizing the difference of squares, leading to the result: \(x(5y - 2)(5y + 2)\). Understanding the underlying structure of algebraic expressions enables mastery of these concepts, enhancing mathematical fluency.