Simplify (x^2-5x)/(x^2-3x)*(x+3)/(x-5)
The given problem involves simplifying a complex algebraic expression. The expression contains two fractions multiplied by each other, each consisting of a polynomial in the numerator and denominator. The task is to perform algebraic manipulations, such as factoring, canceling common factors, and reducing the expression to its simplest form. It's a standard exercise in algebra to practice simplifying rational expressions.
$\frac{x^{2} - 5 x}{x^{2} - 3 x} \cdot \frac{x + 3}{x - 5}$
Answer
Solution:
Step 1: Extract the common factor $x$ from the numerator $x^2 - 5x$.
Step 1.1: Take $x$ out of $x^2$ to get $x(x - 5)$.
Step 1.2: Rewrite the expression as $\frac{x(x - 5)}{x^2 - 3x} \cdot \frac{x + 3}{x - 5}$.
Step 2: Extract the common factor $x$ from the denominator $x^2 - 3x$.
Step 2.1: Take $x$ out of $x^2$ to get $x(x - 3)$.
Step 2.2: Rewrite the expression as $\frac{x(x - 5)}{x(x - 3)} \cdot \frac{x + 3}{x - 5}$.
Step 3: Eliminate the common term $x - 5$.
Step 3.1: Recognize $x - 5$ as a common term in the numerator.
Step 3.2: Cancel out $x - 5$ from both the numerator and denominator.
Step 3.3: Simplify to $\frac{x}{x(x - 3)} \cdot (x + 3)$.
Step 4: Cancel out the common factor $x$.
Step 4.1: Remove the common $x$ from the numerator and denominator.
Step 4.2: Simplify to $\frac{1}{x - 3} \cdot (x + 3)$.
Step 5: Multiply the remaining terms.
- Step 5: Multiply $\frac{1}{x - 3}$ by $x + 3$ to get the final result $\frac{x + 3}{x - 3}$.
Knowledge Notes:
To simplify a rational expression, we can follow these steps:
Factorization: Break down the polynomials in the numerator and the denominator into their factors. This often involves taking out common factors or using other factoring techniques such as factoring quadratics, difference of squares, or sum/product of cubes.
Cancel Common Factors: Once the expression is factored, cancel out any common factors that appear in both the numerator and the denominator. Be careful not to cancel terms that are not factors of the entire polynomial.
Multiplication and Division of Fractions: When multiplying fractions, multiply the numerators together and the denominators together. When dividing, multiply by the reciprocal of the divisor.
Simplification: After canceling and performing the necessary operations, simplify the expression to its lowest terms.
Restrictions: When simplifying rational expressions, it's important to note any restrictions on the variable, which come from values that would make the original denominator equal to zero. These values are not part of the domain of the original expression.
In the given problem, we used these principles to simplify the rational expression by factoring out common terms and canceling them, resulting in a simpler expression.
