Question

Which of the following is an equivalent form of \(\frac{(7 x-7)(7 x+7)}{7}\) ? (A) \(x^{2}-1\) B) \(49 x^{2}+7\) (C) \(7\left(x^{2}-1\right)\) D) \(\frac{\left(x^{2}-7\right)}{7}\)

Short answer

The equivalent form is (C) \(7(x^2 - 1)\).

Step-by-step solution

Simplify the Expression

Start by looking at the expression \(\frac{(7x-7)(7x+7)}{7}\). Notice that the expression inside the numerator can be treated as a difference of squares. Apply the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\). Here, \(a = 7x\) and \(b = 7\), so \(a^2 - b^2 = (7x)^2 - 7^2 = 49x^2 - 49\).

Rewrite Using Difference of Squares

The expression \(\frac{(7x-7)(7x+7)}{7}\) can be rewritten using the difference of squares as \(\frac{49x^2 - 49}{7}\).

Divide Each Term By 7

Divide each term in the numerator by 7: \(\frac{49x^2}{7} - \frac{49}{7}\). This simplifies to \(7x^2 - 7\).

Factor the Resulting Expression

Factor out the common factor of 7 from the expression \(7x^2 - 7\). This gives \(7(x^2 - 1)\).

Compare with the Given Options

The equivalent expression \(7(x^2 - 1)\) matches option (C).

Key concepts

Difference of Squares

The concept of the difference of squares is a crucial technique in algebra that helps in breaking down expressions into more manageable forms. It is based on the formula \(a^2 - b^2 = (a-b)(a+b)\). This formula states that the difference between two squared terms can be expressed as the product of the sum and the difference of their square roots.
In the exercise, we are dealing with the expression \((7x-7)(7x+7)\). Here, by recognizing it as a difference of squares, we can identify \(a = 7x\) and \(b = 7\). Therefore, applying the formula, the expression equals \((7x)^2 - 7^2\), simplifying further to \(49x^2 - 49\).
Why is this important?

• It allows us to condense expressions quickly and efficiently.
• Simplifies the task of factoring and makes it easier to handle fractions.
• Transforms complex expressions into recognizable forms.

Factoring

Factoring involves breaking down an expression into products of simpler expressions or factors. It plays a crucial role in simplifying algebraic expressions or solving equations.
After identifying the numerator \(49x^2 - 49\) of the original expression as a product of square differences, the next step is to further simplify it by factoring. The two terms, \(49x^2\) and \(-49\), have a common factor of 7. By factoring this out, we simplify the expression to \(7(x^2 - 1)\).
Steps to efficient factoring:

• Search for common factors in all terms of the expression.
• Factor them out to reduce the expression to its simplest form.
• Reorganize the expression as necessary to uncover further simplifications.

Simplifying Expressions

Simplifying expressions is the process of reducing algebraic expressions to their most basic form. This is done by applying algebraic operations and properties.In the given exercise, the expression \(\frac{(7x-7)(7x+7)}{7}\) is simplified to \(7(x^2 - 1)\) by successive steps of recognizing patterns and factoring. Initially, the expression was cumbersome and more challenging to interpret. However, by applying the difference of squares and factoring, it becomes far simpler and easier to understand.
Key points to remember when simplifying:

• Identify and cancel common factors in fractions whenever possible.
• Use pattern recognition (like the difference of squares) to facilitate simplification.
• Factor out common terms to reduce complexity and clarify the expression.