Problem 50 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 4 x^{2}-25 $$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Difference of Squares

The expression \( 4x^2 - 25 \) is a difference of squares. This can be identified because it follows the general pattern \( a^2 - b^2 \), where both terms are perfect squares.

Step 2: Identify the Perfect Squares

Notice that \( 4x^2 \) is a perfect square, which is \((2x)^2\), and \( 25 \) is also a perfect square, which is \(5^2\).

Step 3: Apply the Difference of Squares Formula

Use the formula for the difference of squares, \( a^2 - b^2 = (a - b)(a + b) \), where \( a = 2x \) and \( b = 5 \).

Step 4: Write the Factored Form

Substitute \( a \) and \( b \) into the formula: \((2x - 5)(2x + 5)\).

Step 5: Verify the Factorization

Expand \((2x - 5)(2x + 5)\) to check if it equals the original expression: \( (2x)(2x) + (2x)(5) - (5)(2x) - (5)(5) = 4x^2 - 25 \). This confirms the factorization is correct.

Key Concepts

Difference of Squares

Polynomials sometimes take the form of a difference of squares, a concept that simplifies the factoring process considerably. In the case of the polynomial \(4x^2 - 25\), it is identified as a difference of squares. Here's why: the expression is structured as \(a^2 - b^2\), a recognizable theme in algebra where both terms are perfect squares. The hallmark of this structure enables a straightforward factorization using a specific formula.

• Notice 'difference' signifies subtraction between two terms.
• 'Squares' implies each term is the square of another number or expression.

Applying the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\) involves replacing \(a\) and \(b\) with the square roots of the original squared terms. This formula facilitates simple factorization, breaking down complex expressions into manageable pieces.

Perfect Squares

Recognizing perfect squares within a polynomial expression paves the way to using the difference of squares method. For the polynomial \(4x^2 - 25\), identify each term as a perfect square. The term \(4x^2\) can be expressed as \((2x)^2\), and the term \(25\) can be rewritten as \(5^2\).

• Perfect squares are numbers or expressions that can be squared to yield another number.
• They simplify factorization by allowing simple substitution in factoring formulas.

Recognizing both terms as perfect squares is crucial in identifying the type of expression and applying the correct mathematical operations. For instance, without recognizing \(4x^2\) as \((2x)^2\), the factoring process becomes complicated. Spotting these squares early aids immensely in simplifying and solving algebraic expressions.

Factored Form

The goal of factoring a polynomial is to express it in its factored form. This is a productive and powerful representation, often revealing roots and simplified structures. With the difference of squares, the polynomial \(4x^2 - 25\) factors to \((2x - 5)(2x + 5)\).

• The factored form breaks an expression into a product of simpler expressions.
• Recognizing this form can help solve equations and simplify expressions when required.

Verifying the factored form involves expanding it back to check if it matches the original expression. After expansion: \((2x - 5)(2x + 5) = 4x^2 - 25\). Doing this confirms that the factorization is accurate. Factored form is essential both for simplifying algebraic equations and is often the desired goal in problems involving higher degree polynomials.