Problem 51 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 49-4 y^{2} $$

Step-by-Step Solution

Answer

The expression completely factors as \((7 - 2y)(7 + 2y)\).

Step 1: Recognize the Difference of Squares

The expression \(49 - 4y^2\) is a difference of squares. A difference of squares has the general form \(a^2 - b^2\), which factors into \((a-b)(a+b)\). Here, \(49\) is a perfect square since \(49 = 7^2\), and \(4y^2\) is a perfect square since \(4y^2 = (2y)^2\).

Step 2: Identify \(a\) and \(b\)

From the difference of squares formula, identify \(a\) and \(b\). In this expression, \(a = 7\) (since \(7^2 = 49\)) and \(b = 2y\) (since \((2y)^2 = 4y^2\)).

Step 3: Apply the Difference of Squares Formula

Applying the formula for the difference of squares, we substitute \(a = 7\) and \(b = 2y\) into \((a-b)(a+b)\). This results in \((7 - 2y)(7 + 2y)\).

Step 4: Verify the Factorization

To ensure the factorization is correct, expand \((7 - 2y)(7 + 2y)\) to check it equals the original expression.\[(7 - 2y)(7 + 2y) = 7^2 + 2y \cdot 7 - 2y \cdot 7 - (2y)^2 = 49 - 4y^2\] This verifies that \((7 - 2y)(7 + 2y)\) is the correct factorization.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a special technique used to factor certain polynomial expressions. When you have an expression of the form \(a^2 - b^2\), you are dealing with a difference of squares. The magic here is that this can be factored into \((a-b)(a+b)\).
It's helpful to remember some common cases:

• Recognize when you have exactly two terms.
• Check that both terms are perfect squares.
• Ensure there is a minus sign between them—this signals a subtraction or 'difference'.

In the expression \(49 - 4y^2\), \(49\) is \(7^2\) and \(4y^2\) is \((2y)^2\). Spotting these allows you to factor the expression easily.
Understanding these patterns and practicing recognition are key skills in mastering the difference of squares.

Perfect Squares

Perfect squares appear frequently when factoring polynomials, especially when utilizing the difference of squares method. A perfect square is simply a number or expression that can be written as some other number or expression times itself, like \(a^2\).
Common perfect squares include:

• Numbers like 1 (\(1^2\)), 4 (\(2^2\)), 9 (\(3^2\)), and 49 (\(7^2\)).
• Expressions like \(x^2\), \(4x^2 = (2x)^2\), and \(16x^4 = (4x^2)^2\).

Understanding perfect squares is crucial since spotting them allows for easier manipulation and factorization. In our exercise, recognizing both \(49\) and \(4y^2\) as perfect squares was essential to factorize the original expression efficiently.

Polynomial Expressions

Polynomial expressions involve sums and differences of terms, where each term includes numbers, variables raised to powers, or products of numbers and variables. Commonly, they look like \(ax^n + bx^{n-1} + \ldots + k\), where each letter represents coefficients and exponents.
Two key types of polynomial expressions are:

• The 'difference of squares', where the expression has two terms, each a perfect square, separated by a minus sign.
• The 'perfect square trinomial', though not present in this exercise, is a reminder of a closely related form \((a+b)^2\ or (a-b)^2\).

Grasping the structure of polynomial expressions—including recognizing when they can be factored using special patterns—will enhance your algebra proficiency, helping you solve more complex problems in an organized and strategic way. Throughout this article, we've seen how understanding polynomial forms like the difference of squares aids in smooth factorization.