Problem 33 - College Algebra

Question

In Exercises \(31-40,\) factor the difference of two squares. $$36 x^{2}-49$$

Step-by-Step Solution

Answer

So, the factorization of \(36 x^{2} -49\) is \( (6x - 7)(6x + 7) \).

Step 1: Identify \(a\) and \(b\)

First, observe that \(36 x^{2}-49\) can be re-written as \( (6x)^{2} -(7)^{2} \) which fit the pattern \(a^{2}-b^{2}\). Thus, one can see that \(a=6x\) and \(b=7\).

Step 2: Apply the Difference of Squares Formula

With \(a=6x\) and \(b=7\), the formula becomes \( (6x - 7)(6x + 7) \).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic expression. It involves two perfect squares separated by a minus sign. It can be recognized because it follows the form \(a^2 - b^2\). An easy way to spot it is by searching for expressions where both terms are squares. For instance, in \(36x^2 - 49\), we can see that \(36x^2\) is \((6x)^2\) and \(49\) is \(7^2\).
The beauty of the difference of squares is that it can be factored using a straightforward formula: \(a^2 - b^2 = (a - b)(a + b)\). This is because those products simplify the middle terms, leaving only the square components and the difference. This approach rapidly simplifies expressions and is very handy in solving algebraic equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve numbers, variables, and operations. They do not contain an equality sign, which means they represent values not equations. In our example, \(36x^2 - 49\), this is an algebraic expression consisting of two terms. One term is \(36x^2\), a product of a coefficient \(36\) and a variable \(x\) squared. The other term is \(49\), a constant.
Operations involved in algebraic expressions include addition, subtraction, multiplication, and division. In the difference of squares, subtraction is the key operation, separating two squared terms. Understanding the structure of an algebraic expression, like coefficients, variables, and constants, is crucial. These components allow us to manipulate and factor expressions efficiently, like our example with difference of squares.

Polynomial Identities

Polynomial identities are equations that are true for all values of the variables within them. They are tools that allow us to simplify and factor polynomial expressions easily. One crucial identity is the difference of squares identity \(a^2 - b^2 = (a - b)(a + b)\).
This identity provides a quick method to factor polynomials. When faced with a difference of squares, recognizing it allows us to factor it directly using its identity. For instance, in \(36x^2 - 49\), we used the identity to rewrite it as \((6x - 7)(6x + 7)\). This factorization is much simpler to work with, particularly when solving equations or simplifying expressions. Polynomial identities help streamline complex algebraic manipulations by providing proven and efficient strategies.