Problem 24 - College Algebra

Question

For the following exercises, factor the polynomial. $$ 25 y^{2}-196 $$

Step-by-Step Solution

Answer

The factorization is \((5y - 14)(5y + 14)\).

Step 1: Identify the Polynomial Type

The given polynomial is \(25y^2 - 196\). This expression is a difference of squares because both terms can be expressed as squares.

Step 2: Rewrite as a Difference of Squares

Recognize that \(25y^2\) is \((5y)^2\) and \(196\) is \(14^2\). So, the expression can be rewritten as \((5y)^2 - 14^2\).

Step 3: Apply the Difference of Squares Formula

The difference of squares formula states that \(a^2 - b^2 = (a-b)(a+b)\). Here, \(a = 5y\) and \(b = 14\). Applying the formula gives \((5y - 14)(5y + 14)\).

Step 4: Verify the Factorization

Check the factorization by expanding \((5y - 14)(5y + 14)\) using the distributive property: \((5y - 14)(5y + 14) = (5y)^2 + 5y \times 14 - 14 \times 5y - 14^2 = 25y^2 - 196\). This matches the original expression, confirming the factorization is correct.

Key Concepts

Difference of Squares

The difference of squares is a special pattern in a polynomial. It's recognized when an expression is composed of two square terms with a subtraction sign in-between. For example, the expression \( 25y^2 - 196 \) is a difference of squares. This is because it features the squares \((5y)^2\) and \(14^2\), separated by a minus. Here's why it's important:

• Easy Identification: Spotting difference of squares allows for quicker solutions.
• Simplifies Factoring: This technique breaks down complex polynomials into simpler binomial factors.

The key to using this pattern effectively is recognizing the two squared terms and immediately knowing they can be rewritten as \((a-b)(a+b)\). It's really about transforming a subtraction of squares into multiplication of sums and differences.

Polynomial Factorization

Polynomial factorization is the process of breaking a polynomial down into simpler, non-divisible polynomials, or completing its expression into a product of factors. In simpler terms, it’s making a complex expression look like a comprehensible multiplication.In the exponential example \( 25y^2 - 196 \), recognizing it as a difference of squares allows it to be expressed as \((5y - 14)(5y + 14)\). This factorization process involves:

• Analyzing the polynomial to identify patterns such as difference of squares.
• Applying formulas and identities to rewrite the expression.
• Double-checking by multiplying the factors to ensure they reconstruct the original polynomial.

By mastering factorization, difficult-looking expressions become manageable, paving the way for solving equations, simplifying expressions, and finding roots more efficiently.

Algebraic Expressions

In mathematics, an algebraic expression represents relationships using variables and constants, linked by operations like addition, subtraction, multiplication, and division. Expressions can range from simple, like \(3x + 2\), to complex, as seen in the expression \(25y^2 - 196\).Understanding and manipulating these expressions is the cornerstone of algebra. It involves:

• Identifying terms, variables, coefficients, and constants.
• Applying rules and formulas to simplify or solve them.
• Recognizing specific forms, such as the difference of squares or perfect square trinomials.

In the polynomial \(25y^2 - 196\), identifying it as a difference of squares makes it simpler to work with. Recognizing these concepts allows students to focus on the relationships and make accurate calculations irrespective of the complexity of expressions.