Problem 48 - A Graphical Approach to College Algebra

Question

Find each product. $$[(9 r-s)+2][(9 r-s)-2]$$

Step-by-Step Solution

Answer

Result is \(81r^2 - 18rs + s^2 - 4\).

Step 1: Recognize the Formula for Difference of Squares

The given expression \([(9r - s) + 2][(9r - s) - 2]\) is in the form \((a + b)(a - b) = a^2 - b^2\), where \(a = 9r - s\) and \(b = 2\). Identify this as a difference of squares formula.

Step 2: Apply the Difference of Squares Formula

Apply the formula \((a + b)(a - b) = a^2 - b^2\) to the expression. Here, \(a = 9r - s\) and \(b = 2\). Substitute these values into the formula to get \((9r - s)^2 - 2^2\).

Step 3: Simplify the Expression

Calculate each part of the expression: for \((9r - s)^2\), use \((9r - s)^2 = (9r - s)(9r - s) = 81r^2 - 18rs + s^2\). For \(2^2\), compute \(2^2 = 4\).

Step 4: Write the Final Expression

Subtract the constants: \(81r^2 - 18rs + s^2 - 4\). This is the simplified form of the expression after applying the difference of squares formula.

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental building blocks in mathematics, especially when dealing with equations and formulas. An algebraic expression consists of variables, numbers, and operations combined together to represent a particular value or relationship.

In the context of the difference of squares, the expression \[(9r - s) + 2\]and \[(9r - s) - 2\] are algebraic expressions that can be multiplied together. Each part of an algebraic expression has specific components:

• Terms: These are individual parts within an expression, separated by a plus or minus sign, such as \[9r\] and \[-s\]. In the given problem, terms include \[9r\], \[-s\], and constants like \[2\].
• Coefficients: This is a number that multiplies a variable, for instance, \[9\] is the coefficient of \[r\].
• Variables: These symbols, like \[r\] and \[s\], represent numbers we don't know yet.
• Constants: Fixed values in the expression, such as \[2\].

Understanding these components helps break down and simplify expressions into solvable problems. Recognizing patterns, such as the structure of a difference of squares, allows us to manipulate expressions effectively.

Polynomial Equations

Polynomial equations involve expressions made up of multiple terms where variables have whole number exponents. These kinds of equations appear frequently in algebra and can range in complexity from simple linear equations to complex cubic equations.

In our exercise, when we simplified\[(9r - s)^2 - 4\],we dealt with a polynomial equation resulting from applying the difference of squares.

• Degree: This indicates the highest power of the variable, in our simplified equation, it's \[r^2\], showing it's quadratic.
• Operations: Polynomials contain operations of addition, subtraction, and multiplication between terms. Division by a variable is avoided to maintain polynomial forms.
• Standard form: Polynomials are often written in standard form, with the highest degree term first. Our simplified expression \[81r^2 - 18rs + s^2 - 4\] is an example.

Recognizing polynomial equations lets us apply various algebraic techniques and theorems, leading to simplified solutions or factored forms, especially useful for quadratic versions like this one.

Mathematical Formulas

Mathematical formulas are essential tools in problem-solving, offering general solutions to particular problem types. For example, the difference of squares formula is a powerful shortcut that recognizes an easy-to-solve pattern.

In this exercise, we used the formula for the difference of squares: \[(a + b)(a - b) = a^2 - b^2\]. This straightforward identity simplifies the process of expanding expressions without multiplying each term individually.

• Substitution: Identifying \[a\] and \[b\] in the expression and substituting them into the formula gives a quick solution.
• Application: Using formulas involves recognizing when they apply. Here, the expression followed the \[(a+b)(a-b)\] structure, qualifying it for the difference of squares.
• Simplification: After substitution, further simplifying results in completely managed polynomial expressions, i.e., \[81r^2 - 18rs + s^2 - 4\].

Familiarity with mathematical formulas reduces calculation time and simplifies complex expressions, ensuring quick and efficient problem-solving.