Simplify (x^2-2x)/x
The given problem is a mathematical expression that requires simplification. Specifically, you're provided with a rational expression where a quadratic polynomial, x^2 - 2x, is in the numerator, and a linear polynomial, x, is in the denominator. The task is to simplify this fraction by canceling out any common factors that appear in both the numerator and the denominator. Simplifying such expressions typically results in a more compact form, which may be easier to evaluate or use in further calculations.
$\frac{x^{2} - 2 x}{x}$
Answer
Solution:
Step 1: Extract the common factor from the numerator
Extract the factor $x$ from the terms in the numerator $x^{2} - 2x$.
Step 1.1: Factor out $x$ from $x^{2}$
Write $x^{2}$ as $x \cdot x$. The expression becomes $\frac{x \cdot x - 2x}{x}$.
Step 1.2: Factor out $x$ from $-2x$
Express $-2x$ as $x \cdot (-2)$. The expression now reads $\frac{x \cdot x + x \cdot (-2)}{x}$.
Step 1.3: Combine the factored terms
Combine the factored terms to show the common factor explicitly: $\frac{x(x - 2)}{x}$.
Step 2: Simplify the fraction by canceling out the common factor
Eliminate the common $x$ factor from the numerator and denominator.
Step 2.1: Cancel the $x$ factor
The $x$ in the numerator and denominator cancels out: $\frac{\cancel{x}(x - 2)}{\cancel{x}}$.
Step 2.2: Final simplification
The expression simplifies to $x - 2$ after canceling the common factor.
Knowledge Notes:
To simplify a rational expression like $\frac{x^{2} - 2x}{x}$, we can use the following knowledge points:
Factoring: This involves expressing an algebraic expression as a product of its factors. In this case, we factor out the common term $x$ from the numerator.
Common Factors: When the same factor appears in both the numerator and denominator of a fraction, it can be canceled out, as long as it is not equal to zero. This is based on the property that $\frac{a}{a} = 1$ for any $a \neq 0$.
Simplification: After canceling common factors, the expression is simplified to its lowest terms. This process often involves reducing the number of terms in the expression and making it more straightforward.
Algebraic Division: When we cancel out the common factors, we are effectively dividing the terms by themselves, which results in a value of 1.
Restrictions: It's important to note that when simplifying expressions that involve variables in the denominator, we must consider the restrictions on the variable values. In this case, $x$ cannot be zero because division by zero is undefined.
By applying these principles, we can simplify algebraic expressions and solve various algebraic problems.
