Problem 19 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$(x+2)^{2}-(x+7)^{2}$$

Step-by-Step Solution

Answer

Factored form is \((-5)(2x+9)\).

Step 1: Recognize the Difference of Squares

The expression \((x+2)^{2}-(x+7)^{2}\) is a difference of squares because it is in the form \(a^2 - b^2\), where \(a = x+2\) and \(b = x+7\). The difference of squares pattern is \(a^2 - b^2 = (a - b)(a + b)\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a = x+2\) and \(b = x+7\) into the formula: \[(x+2)^{2}-(x+7)^{2} = ((x+2)-(x+7))((x+2)+(x+7))\]

Step 3: Simplify the Expression

First, simplify \((x+2)-(x+7)\): \((x+2)-(x+7) = x + 2 - x - 7 = -5\)Next, simplify \((x+2)+(x+7)\): \((x+2)+(x+7) = x + 2 + x + 7 = 2x + 9\)

Step 4: Write the Final Factored Expression

Combine the simplified terms from Step 3 to write the final factored expression: \((x+2)^{2}-(x+7)^{2} = (-5)(2x+9)\)

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that simplifies expressions and solves equations more feasible. By breaking down a polynomial into simpler components or factors, you make it easier to understand and work with. In this exercise, we're specifically dealing with the method of factoring using the "difference of squares". This method applies when you have an expression that fits the format of \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\).
This is a special polynomial case where every number or variable squared has a distinct square root, making it notably straightforward to factor.
Here are some key points to remember about factoring polynomials:

• Identifying the pattern in expressions is crucial. Look for the square terms first.
• Factor them by applying appropriate formulas such as the difference of squares.
• Simplifying the resultant factors can help in identifying further common factors, if any exist.

Recognizing these patterns and formulas not only helps with simplifying expressions but also in solving polynomial equations efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They're fundamental in expressing mathematical relationships, and they set the stage for equations.
Expressions like

• \((x+2)\) and \((x+7)\)

are considered simple algebraic expressions. When squared and subtracted from each other, they form the structure of the difference of squares, which is higher-level algebra.
An important aspect of algebraic expressions:

• They're the building blocks for more complex mathematics, such as equations and functions.
• In any subtraction or addition within them, maintain proper attention to detail because every part contributes to the bigger picture in math operations.
• Recognizing where an expression can be modified or simplified is part of mastering algebra.

Understanding algebraic expressions and how they can be factored or simplified is vital for progress in algebra.

Mathematical Formulas

Mathematical formulas serve as standardized methods for solving a wide variety of problems. In the context of this problem, the formula for the difference of squares \(a^2 - b^2 = (a-b)(a+b)\) is instrumental. It provides a straightforward way to decompose specific polynomial expressions.
Formulas are valuable because:

• They save time and reduce the potential for error by providing step-by-step methods.
• They assist in recognizing patterns quickly, which simplifies complex problems.
• Knowing a formula allows you to approach a problem strategically instead of always calculating from scratch.

In algebra, having a bank of formulas at your disposal is immensely beneficial, especially for factoring expressions like the difference of squares.
With practice, applying such formulas becomes intuitive, making them indispensable tools in a student's mathematical journey.