Problem 102 - Algebra and Trigonometry

Question

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{x^{2}-25}{x-5}=x-5 $$

Step-by-Step Solution

Answer

The original expression \(\frac{x^{2}-25}{x-5}=x-5\) is false. However, it can be corrected to a true statement as \(\frac{x^{2}-25}{x-5}=x+5\).

Step 1: Initial Check

First, let's examine the equation as it is. It can't be checked through direct substitution for \(x=5\) because this would result in division by zero. Therefore, more generally, it needs to be simplified. The numerator, \(x^{2}-25\), can be factored using the difference of squares technique.

Step 2: Factor the numerator

The numerator can be factored as \((x-5)(x+5)\) because it follows the \(a^{2}-b^{2} = (a−b)(a+b)\) format. So, the equation becomes: \[\frac{(x-5)(x+5)}{x-5}=x-5\] Now, assuming \(x \neq 5\) to avoid division by zero, we can cancel out the common factor \((x-5)\) on both sides.

Step 3: Simplify and check the simplified expression

After cancelling, the equation simplifies to \(x+5 = x-5\). However, this statement is not true for every value of \(x\), in fact, there is no value of \(x\) that would make this equation hold true. Therefore, the original statement is false.

Step 4: Correct the equation

To correct the equation and make it a true statement, change the right side of the equation to \(x+5\). Now, the equation reads: \[\frac{x^{2}-25}{x-5}=x+5\] This equation is true for any real number \(x\) other than 5.

Key Concepts

Difference of Squares

The "difference of squares" is a formula in algebra that helps us factor certain kinds of expressions. This formula is particularly useful when you are dealing with expressions like \( x^2 - a^2 \). The formula states\[ x^2 - a^2 = (x - a)(x + a) \]This means that a square of one term subtracted from the square of another can be expressed as the product of a difference and a sum.

• This trick is handy for simplifying polynomials.
• It reveals that many polynomials are products of simpler terms.
• In our exercise, \( x^2 - 25 = (x - 5)(x + 5) \).

Therefore, understanding this pattern allows us to break complex expressions into simpler parts.
This can be particularly helpful during polynomial division.

Polynomial Division

Polynomial division is similar to long division with numbers. In this process, we divide a polynomial (the dividend) by another polynomial (the divisor) to get a quotient and possibly a remainder.

• In our example, we divide \((x^2 - 25)\) by \((x - 5)\).
• Once the numerator is factored as \((x - 5)(x + 5)\), you see the common term \((x - 5)\) in both the numerator and the denominator.
• By cancelling these terms, the division simplifies.

After dividing, the equation simplifies to \(x + 5 = x - 5\).
However, unless further conditions or values are specified, this does not hold true for any \(x\).

Division by Zero

Division by zero is one of the critical concepts in algebra that students often encounter. The expression \(\frac{a}{b}\) becomes undefined when \(b = 0\). This is because dividing a number by zero does not yield a meaningful result, and it leads to mathematical inconsistencies.

• In our case, the denominator \((x - 5)\) is zero when \(x = 5\).
• Avoiding division by zero is crucial when simplifying expressions, which is why we exclude \(x = 5\) from solutions.
• This exclusion ensures the operation is defined and valid.

Thus, being aware of such limitations is important, especially while simplifying and evaluating statements.

Simplification in Algebra

Simplification in algebra means reducing expressions into their simplest form. This often involves using factoring, cancelling common terms, and eliminating complexities.

• Simplification helps assess the truth of given equations, just as we did in our example.
• Originally, \(x^2 - 25 = (x - 5)(x + 5)\) allowed us to cancel the \((x - 5)\) common factor.
• Simplifying leads to expressions where comparing terms becomes easier.

Upon simplifying, we observed that \(x + 5 = x - 5\) doesn't hold for any real number \(x\), indicating the need to correct or adjust the statement.
Thus, the final simplified and true statement would read \( \frac{x^2 - 25}{x - 5} = x + 5 \), except at \(x = 5\), where the expression becomes undefined.