Problem 43 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ 81 x^{2}-144 $$

Step-by-Step Solution

Answer

The factors of the expression \(81x^{2} - 144\) are \( (9x - 12) \) and \( (9x + 12) \).

Step 1: Identify the difference of squares

In the given expression \(81x^{2} - 144\), identify that both terms are perfect squares. Here \(81x^{2}\) is the square of \(9x\) and \(144\) is the square of \(12\). Thus, the original equation can be represented as \((9x)^{2} - 12^{2}\)

Step 2: Applying the difference of squares formula

Apply the difference of squares formula, which is \(a^{2}-b^{2} = (a - b)(a + b)\). Hence, \((9x)^{2} - 12^{2} = (9x - 12)(9x + 12)\).

Step 3: Simplification

Simplify the equation \((9x - 12)(9x + 12)\) and we get \(9x - 12\) and \(9x + 12\) as the factors of given expression.

Key Concepts

Perfect Squares

When tackling algebraic expressions, recognizing perfect squares can simplify the process significantly. A perfect square is the product of a number multiplied by itself, such as 4 (which is 2 times 2) or 9 (3 times 3). In the context of the exercise, we encounter the numbers 81 and 144. Both of these numbers are indeed perfect squares: 81 is 9 squared (or 9 times 9), while 144 is 12 squared (12 times 12).

In algebra, we may also encounter variables raised to an even power, like in the term \(81x^{2}\). This signifies that the term is the squared result of \(9x\), making \(9x\) the base. Being able to quickly identify these squares is essential for factoring expressions efficiently, taking us one step closer to simplifying equations or solving algebraic problems.

Algebraic Expressions

Algebraic expressions encompass terms and operations that include variables, coefficients, and constants. In the given exercise, \(81x^{2}-144\) is an algebraic expression that includes the variable \(x\), multiplied by a coefficient (81), and a constant term (144). Understanding algebraic expressions is fundamental because they represent mathematical relationships that can be manipulated through operations such as factoring, multiplying, or simplifying.

Breaking down complex algebraic expressions into simpler components can often unveil patterns — such as the difference of squares in this case — facilitating the process of finding solutions or simplifying the expression further.

Polynomial Factoring

Polynomial factoring is an essential skill in algebra, and it involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, would give the original polynomial. Factoring can greatly aid in solving equations as well as simplifying expressions. In our example, we're dealing with a difference of squares, which is a form of polynomial factoring. Recognizing that both terms in the expression are perfect squares, we can apply a specific factoring rule. This strategy is not only helpful for difference of squares but extends to other factoring techniques such as factoring trinomials or recognizing common factors.

Simplifying Equations

Simplifying equations is about reducing complexity to reveal a more understandable form. It can involve combining like terms, factoring expressions, or canceling out terms. In the given exercise, after recognizing the difference of squares, we apply the appropriate formula and factor the expression into \((9x - 12)(9x + 12)\). This doesn't just simplify the expression; it also paves the way for further algebraic manipulation. For instance, if this were part of an equation set to zero, we could now easily solve for \(x\) by setting each factor equal to zero. Simplifying is not only about making an expression look neater; it allows for greater insight into the properties and potential solutions of the expression.