Problem 25 - College Algebra

Question

For the following exercises, factor the polynomial. $$ 121 p^{2}-169 $$

Step-by-Step Solution

Answer

The factors are \((11p - 13)(11p + 13)\).

Step 1: Identify the Type of Polynomial

The polynomial given is a difference of squares because it can be expressed as the difference between two perfect squares: \( 121 p^2 - 169 = (11p)^2 - 13^2 \). This is a special form of polynomial known as a difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is given by \[ a^2 - b^2 = (a - b)(a + b) \]. In our case, \( a = 11p \) and \( b = 13 \). Therefore, \[ 121 p^2 - 169 = (11p)^2 - 13^2 = (11p - 13)(11p + 13) \].

Key Concepts

Difference of Squares

Understanding the concept of a difference of squares is crucial because it simplifies factoring certain types of polynomial expressions. A difference of squares refers to a specific pattern where two perfect square terms are subtracted. The general form is expressed as \( a^2 - b^2 \). When you see this pattern, you can immediately factor it using the formula:

• \( a^2 - b^2 = (a - b)(a + b) \)

This formula works universally for any difference of squares expression you encounter.
Recognizing and applying this pattern can greatly reduce the work involved in factoring since it directly gives you the binomial factors of the expression.

Polynomial Expressions

Polynomials are algebraic expressions that consist of variables raised to whole number powers and coefficients. They can have one or more terms, where each term is a part of the whole polynomial, such as \( 121p^2 \) or \( -169 \). A term is typically made up of:

• a coefficient (numerical factor such as 121 or -169),
• and a variable raised to an exponent (like \( p^2 \)).

Knowing how to identify and manipulate these parts helps in simplifying and factoring polynomial expressions.
In our example, \( 121p^2 - 169 \), each term is a perfect square—something you look for when applying the difference of squares rule.Recognizing that each term is squared allows you to quickly identify appropriate factoring methods such as the difference of squares.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve problems efficiently. When factoring polynomials, it's important to follow specific steps to transform expressions correctly.

• Identify any common patterns or formulas, such as the difference of squares, to simplify the process.
• Express terms as squares to fit those patterns.

In our exercise, starting with identifying \( 121p^2 - 169 \) as a difference of squares was crucial. Once recognized, you can use algebraic manipulation to transform it using the formula:

• \( (11p)^2 - 13^2 = (11p - 13)(11p + 13) \)

This type of manipulation allows for simplifying complex expressions into easier, factorized forms, aiding in both solving and understanding polynomial problems efficiently.