Problem 62 - Elementary Algebra

Question

For the following problems, find the products. $$ \left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right) $$

Step-by-Step Solution

Answer

Question: Evaluate the product of the given expressions: \(\left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right)\) Answer: \(x^2 - \frac{49}{9}\)

Step 1: Identify the values of \(a\) and \(b\)

Here, we have: $$a = x$$ $$b = \frac{7}{3}$$

Step 2: Apply the formula for the product of a sum and a difference of two terms

We can now apply the formula to find the product: $$\left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right) = \left(a+b\right)\left(a-b\right) = a^2 - b^2$$

Step 3: Substitute the values of \(a\) and \(b\) and simplify

Now, substitute the values of \(a\) and \(b\) into the formula and simplify. We get: $$\left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right) = x^2 - \left(\frac{7}{3}\right)^2$$ Calculating the square of \(\frac{7}{3}\): $$\left(\frac{7}{3}\right)^2 = \frac{49}{9}$$

Step 4: Write the final answer

The final answer is: $$\left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right) = x^2 - \frac{49}{9}$$

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves breaking down an expression into a product of simpler expressions or numbers. Think of it as reverse-multiplication. Instead of multiplying expressions, we are trying to find what was multiplied together to create the original expression.

When we factor polynomials, we are essentially looking for terms that can be multiplied together to form the polynomial. For example, the expression \( x^2 - \frac{49}{9} \) can be factored into \( (x + \frac{7}{3})(x - \frac{7}{3}) \).

• It is important to identify any common factors in the terms of the polynomial and use distributive properties if applicable.
• Basic factoring methods involve removing a greatest common factor and grouping like terms.

Factoring skills are essential for simplifying algebraic expressions and solving polynomial equations. Mastery of this skill will make solving such expressions much smoother and more intuitive.

Difference of Squares

The difference of squares is a special pattern used in algebra where two squared terms are subtracted from one another. It appears in the form \( a^2 - b^2 \) and can be factored using the formula \( (a + b)(a - b) \). This pattern is incredibly useful for simplifying and factoring polynomials.

• Recognizing this pattern can help you quickly factor expressions without having to multiply everything out.
• The difference of squares takes advantage of the identities \( a^2 - b^2 = (a+b)(a-b) \).

In the solution of the exercise, the expression \( \left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right) \) uses this property to simplify \( a^2 - b^2 \), resulting in \( x^2 - \frac{49}{9} \). This makes it easier to work with and further analyze polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators that represent a specific value or set of values. They are fundamental in algebra since they form the basis of equations and functions.

• Expressions can vary greatly in complexity, ranging from a simple \( x + 3 \) to more intricate ones like \( \left(x+\frac{7}{3}\right)\left(x-\frac{7}{3}\right) \).
• Understanding how to manipulate algebraic expressions is key to mastering algebraic operations and solving equations.

Algebraic expressions can include:

• Variables – symbols that represent numbers whose values can change.
• Constants – fixed values that do not change.
• Operators – symbols that indicate operations like addition, subtraction, multiplication, and division.

By learning to recognize and work with algebraic expressions, students can better grasp how to isolate variables and solve equations. This knowledge allows for a deeper understanding of mathematical concepts and enhances problem-solving capabilities.