Problem 66 - Algebra A Combined Function

Question

Simplify each expression. Then determine whether the given answer is correct. $$ \frac{100-x^{2}}{x-10} ; \text { Answer: }-10-x $$

Step-by-Step Solution

Answer

The given answer \(-10-x\) is correct after simplification.

Step 1: Identify the Algebraic Expression

The expression given is \( \frac{100-x^2}{x-10} \). We need to simplify this expression.

Step 2: Recognize a Difference of Squares

Observe that the numerator \( 100 - x^2 \) is a difference of squares. It can be factored as \( (10-x)(10+x) \).

Step 3: Factor the Expression

Write the expression as \( \frac{(10-x)(10+x)}{x-10} \).

Step 4: Simplify the Fraction

Since \( (10-x) \) is equivalent to \( -(x-10) \), rewrite the expression: \[ \frac{-(x-10)(10+x)}{x-10} \].

Step 5: Cancel Common Factors

Cancel the \( x-10 \) from the numerator and the denominator: \( -(10+x) \).

Step 6: Verify the Simplified Expression

The simplified expression is \( -10 - x \), which matches the answer provided: \( -10 - x \). Therefore, the given answer is correct.

Key Concepts

Difference of Squares

When you encounter expressions like \(100 - x^2\), it's essential to recognize the pattern known as the "Difference of Squares." This pattern occurs when you have two terms squared and subtracted from each other. The formula is generally written as \(a^2 - b^2\). This expression can always be factored into \((a-b)(a+b)\).
This is a fundamental concept in algebra that allows for easier manipulation and simplification of expressions. For example, in our exercise, \(100\) is \(10^2\) and \(x^2\) remains as is. Therefore, applying the difference of squares, \(100 - x^2\) becomes \((10-x)(10+x)\). Understanding how to quickly spot and factor these expressions is a key skill in algebra that can be used in various applications, such as simplifying fractions or solving equations.

Factoring

Factoring is a method used to break down expressions into simpler components that, when multiplied together, yield the original expression. It is crucial for simplifying algebraic expressions, solving equations, and understanding polynomial identities.
In the given exercise, after recognizing the numerator \(100-x^2\) as a difference of squares, we factor it into \((10-x)(10+x)\). This is an essential step to simplifying the fraction. By factoring, you prepare the expression for further simplification, such as canceling out common terms. It transforms complex expressions into a product of simpler expressions, making calculations more manageable and less error-prone.
If you become skilled at factoring, you will find many algebraic problems become easier and quicker to solve.

Algebraic Fractions

Algebraic fractions are expressions that involve variables in the numerator, denominator, or both. Simplifying these fractions often requires techniques like factoring and recognizing special patterns like the difference of squares.
In the given exercise, we have an algebraic fraction \(\frac{100-x^2}{x-10}\). The aim is to simplify this fraction, which often means reducing it to the lowest terms or a simpler form. After factoring the numerator, the expression becomes \(\frac{(10-x)(10+x)}{x-10}\).
To simplify an algebraic fraction effectively, always look to cancel common factors from the numerator and the denominator, assuming they are not zero. Here, noticing that \((10-x)\) can be rewritten as \(-(x-10)\) allows the \(x-10\) in the denominator and numerator to cancel each other, simplifying the expression to \(-(10 + x)\). Thus, learning to manipulate, factor, and simplify algebraic fractions is crucial for tackling a wide range of algebraic problems efficiently.