Problem 28 - Elementary Algebra

Question

For the following problems, factor the binomials. $$ 121 a^{2}-9 $$

Step-by-Step Solution

Answer

Answer: The factored form of the expression $$121a^2 - 9$$ is $$(11a - 3)(11a + 3)$$.

Step 1: Identify the Difference of Squares

Recognize the given expression as the difference of two squares: $$(11a)^2 - (3)^2$$.

Step 2: Apply the Difference of Squares Formula

We will use the formula $$(a^2 - b^2) = (a - b)(a + b)$$. In our case, let \(a = 11a\) and \(b = 3\). So, the expression becomes: $$(11a - 3)(11a + 3)$$.

Step 3: Write the Final Factored Expression

The factored expression is: $$(11a - 3)(11a + 3)$$.

Key Concepts

Difference of Squares

When we talk about the difference of squares, we're referring to a special factoring scenario in algebra. This is a situation where two squared terms are subtracted from one another. The general pattern for recognizing a difference of squares is when you have an expression like \(a^2 - b^2\). This pattern is special because it can always be factored into \((a - b)(a + b)\).
The expression \(121a^2 - 9\) fits this pattern perfectly. Here, \(121a^2\) is the square of \(11a\), since \((11a)^2 = 121a^2\). Similarly, \(9\) is the square of \(3\), because \(3^2 = 9\). Recognizing these components allows us to rewrite the expression as \((11a)^2 - 3^2\). This is what we mean when we say we're identifying a difference of squares. Once identified, it can be factored neatly into two binomials: \((11a - 3)(11a + 3)\).
Understanding the difference of squares can greatly simplify problems that might at first seem complicated. It's an efficient technique that often appears in algebra.

Factoring Methods

Factoring is an essential algebraic skill that allows us to simplify expressions and solve equations more easily. Several methods can be used for factoring polynomials, and each is suitable for different kinds of expressions.

• Common Factoring: Look for a common factor in all the terms. This is the simplest factoring approach.
• Difference of Squares: Used specifically when an expression is in the form of \(a^2 - b^2\), as illustrated above.
• Trinomials: Factor by finding two binomials that multiply to give the original trinomial.
• Grouping: Useful for expressions with four or more terms. It involves grouping terms to reveal common factors.

When tackling the exercise \(121a^2 - 9\), we chose the difference of squares method because the expression fits the pattern perfectly. Identifying the best factoring method is crucial, as it often saves time and simplifies the process.

Polynomial Expressions

Polynomials are expressions made with numbers and variables using the operations of addition, subtraction, and multiplication. A polynomial can have constants, variables, and exponents that are non-negative integers.
A "binomial" is a polynomial with exactly two terms. In our exercise, \(121a^2 - 9\) is a classic example of a binomial expression because it has two terms: \(121a^2\) and \(-9\).
Polynomials play a vital role in mathematics, serving as a foundation for numerous algebraic concepts. Understanding how to manipulate and factor them allows us to simplify many mathematical problems and find solutions to algebraic equations.
In factoring, we're often attempting to rewrite a polynomial as a product of simpler polynomials. This is crucial because it can make solving equations easier and reveals important properties of the polynomial.