Problem 4 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$4 x^{2}-49$$

Step-by-Step Solution

Answer

The factored form is \((2x - 7)(2x + 7)\).

Step 1: Recognize the Pattern

The difference-of-squares pattern follows the rule \(a^2 - b^2 = (a - b)(a + b)\). Here, we need to express both terms as squares.

Step 2: Rewrite Each Term as a Square

Express the first term \(4x^2\) as \((2x)^2\) and the second term \(49\) as \(7^2\). This gives us the expression \((2x)^2 - (7)^2\).

Step 3: Apply the Difference of Squares Formula

Using the rule \(a^2 - b^2 = (a - b)(a + b)\), we substitute \(a = 2x\) and \(b = 7\). This yields \((2x - 7)(2x + 7)\).

Step 4: Verify the Factorization

To ensure correctness, expand \((2x - 7)(2x + 7)\) back to the original expression. \((2x - 7)(2x + 7) = (2x)^2 - (7)^2 = 4x^2 - 49\), which matches the original expression.

Key Concepts

Factoring Polynomials

Factoring polynomials might seem daunting at first, but with a bit of practice, it becomes more straightforward and intuitive. One of the patterns frequently encountered in this process is the "difference of squares pattern." This allows you to break down specific polynomial expressions into simpler ones, making them easier to handle or solve.

• To begin factoring, always look for common patterns or factors shared by the terms.
• In the case of difference of squares, you look for expressions that can be written in the form of \(a^2 - b^2\).
• Once identified, you can factor it as \((a - b)(a + b)\), which simplifies the expression significantly.

Factoring not only simplifies the expression but also helps in solving polynomial equations by allowing you to find the roots quickly. Practice identifying different patterns and applying them—your polynomials will transform before your very eyes!
Remember, practice is crucial in mastering these techniques.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. These can range from simple expressions like \(x + 2\) to more complex ones involving multiple terms, exponents, and variables.

• Each term in an algebraic expression can be a number (constant), a variable, or a product of numbers and variables.
• Operations involved include addition, subtraction, multiplication, and division.

In the exercise, we dealt with the algebraic expression \(4x^2 - 49\). Recognizing that these terms can be re-written in a specific form, using their square roots, is pivotal. This recognition is the first step in leveraging the "difference of squares" strategy, making complex expressions easier to deal with.
Develop a habit of consistently breaking down terms, seeing past simple surface level, to their deeper algebraic forms.

Polynomial Identities

Polynomial identities are equations that hold true for any value of the variable used. They often represent special cases where an equation simplifies consistently into the same form.

• The difference of squares is a polynomial identity: \(a^2 - b^2 = (a-b)(a+b)\).
• Recognizing these identities helps in both simplifying calculations and understanding general algebraic manipulation.

In our example, identifying \((2x - 7)(2x + 7)\) as the expanded form of \(4x^2 - 49\) showcases the importance of understanding these identities.
Mastering polynomial identities like these will not only improve your algebra skills but also lay a solid foundation for more advanced mathematics, as they provide a powerful tool for factoring and simplifying expressions.