Problem 46 - A Graphical Approach to College Algebra

Question

Factor each difference of squares completely. $$(p-2 q)^{2}-100$$

Step-by-Step Solution

Answer

The factored form is \((p - 2q - 10)(p - 2q + 10)\).

Step 1: Identify the Difference of Squares

The expression given is \((p-2q)^2 - 100\). Notice that this is a difference of two squares, where \((p-2q)^2\) is the first square and \(100\) is the second square, since \(100 = 10^2\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a-b)(a+b)\). Here, \(a = (p-2q)\) and \(b = 10\). Substitute these values into the formula to get: \[(p-2q)^2 - 10^2 = ((p-2q) - 10)((p-2q) + 10)\].

Step 3: Simplify the Factored Expression

Simplify each binomial in the factored expression. - For \((p-2q) - 10\), simply subtract 10 from \(p-2q\) to get \(p-2q-10\).- For \((p-2q) + 10\), add 10 to \(p-2q\) to get \(p-2q+10\).Thus, the fully factored expression is \((p - 2q - 10)(p - 2q + 10)\).

Key Concepts

Factoring

Factoring is a fundamental concept in algebra, where we break down complex expressions into simpler component part---factors.
This helps in making equations easier to solve and manipulate. Factoring identities, like the difference of squares, are tools that simplify the process, especially when dealing with quadratic expressions or polynomials.

• In our exercise, we are looking at the difference of squares, which is a special case of factoring.
• A difference of squares has the form \(a^2 - b^2\) and can be factored as \((a-b)(a+b)\).
These formulas emphasize the relationship between numbers or expressions, acting as shortcuts that bypass the need for more complex calculations. By correctly identifying and applying these shortcuts, we can efficiently simplify and solve mathematical problems. This method not only aids in breaking down expressions but can often uncover hidden relationships between the elements of an expression.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations.
They form the backbone of algebra and by extension, higher-level math. Mathematical operations such as addition, subtraction, multiplication, and division are used to combine these variables and numbers in meaningful ways.

• The expression in our exercise is \((p-2q)^2 - 100\). It combines a binomial \((p-2q)\) with exponentiation and a constant \(-100\).
• Our task is to manipulate the expression through factoring, revealing its simplest form via identifiable mathematical patterns.

These components within algebraic expressions allow for modeling real-world situations, providing a way to express and solve problems algebraically. Understanding how these components interact is key to mastering algebraic manipulation and simplification.

Binomials

In algebra, a binomial is an expression containing exactly two terms, such as \((p - 2q)\).
Binomials are significant because they often serve as building blocks for polynomial expressions.

• In our exercise, the binomial \((p - 2q)\) is squared, resulting in \((p - 2q)^2\).
• This serves as one part of the difference of squares formula used to factor the overall expression.

Binomials are manipulated using principles such as distribution and factoring. By understanding the properties of binomials, one can simplify and solve algebraic expressions effectively. Recognizing patterns involving binomials, like the difference of squares, enhances algebraic fluency and problem-solving skills.