Problem 37 - Beginning and Intermediate Algebra

Question

Factor completely. $$n^{2}-121$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial is \((n + 11)(n - 11)\).

Step 1: Identify the difference of squares

Here we have a polynomial in the form \(n^2 - 121\). Notice that both the terms are perfect squares: \(n^2 = n*n\) and \(121 = 11*11\). Thus, our polynomial can be written as a difference of squares, which takes the form \((a^2 - b^2)\). In our case, \(a = n\) and \(b = 11\).

Step 2: Factor using the difference of squares formula

To factor a difference of squares, we'll use the formula \((a^2 - b^2) = (a+b)(a-b)\). Plugging in our values for a and b, we get: \((n^2 - 121) = (n+11)(n-11)\). So, the completely factored form of the given polynomial is \((n + 11)(n - 11)\).

Key Concepts

difference of squares

The concept of the "difference of squares" is a valuable tool in algebra for factoring certain types of polynomials. Let's break this down. A difference of squares exists when a polynomial can be expressed in the form \(a^2 - b^2\), where both \(a^2\) and \(b^2\) are perfect squares. This is a structure we can exploit because it simplifies into two binomials:

• \((a + b)(a - b)\)

This means if you know a polynomial fits this pattern, you can immediately use this formula. In our exercise, \(n^2 - 121\), both \(n^2\) and \(121\) are perfect squares, where \(a = n\) and \(b = 11\). Therefore,

• \(n^2 - 121 = (n+11)(n-11)\)

Recognizing and applying the difference of squares allows for quick and efficient factoring.

polynomials

A polynomial is a mathematical expression involving a sum of powers of variables with coefficients. They can appear intimidating at first, but they are foundational to many algebraic operations and problem-solving techniques. A polynomial can be as simple as \(x + 2\) or as complex as \(4x^3 + 3x^2 - x + 6\). In the realm of factoring, our goal is often to break down these expressions into their simplest parts, much like breaking down a sentence into words. When dealing with a quadratic polynomial like \(n^2 - 121\), it often helps to look for structures or patterns such as factored forms, which can lead to simplification. Recognizing these patterns helps massively in calculus, physics, and engineering, where polynomials frequently model real-world phenomena.

perfect squares

Perfect squares are numbers or expressions formed by squaring an integer or a variable, such as 1, 4, 9, 16, etc., or \(x^2, (x+1)^2\), and so forth. Identifying perfect squares is crucial in the factoring process, especially in problems involving the difference of squares.Consider the number 121; it's a perfect square because it is equal to \(11^2\). Similarly, \(n^2\) is a perfect square because it equals \(n\cdot n\). Recognizing that both components in \(n^2 - 121\) are perfect squares is the key step that allows us to apply the difference of squares formula and factor the expression completely.

• Perfect squares often appear in patterns: \((a^2 - b^2), (a + b)^2, (a - b)^2\).
• They help simplify complex algebraic expressions by spotting these patterns early on in the computation process.

Being comfortable with perfect squares can thus make a significant difference in the ease with which you can approach and solve algebraic problems.