%2F(x%2B4).jpeg "(x^2-3x-33)/(x+4)")
Simplifying Rational Expressions: (x^2 - 3x - 33) / (x + 4)
In mathematics, a rational expression is a fraction where the numerator and denominator are both polynomials. Simplifying rational expressions involves reducing them to their simplest form. This involves factoring the numerator and denominator and then canceling out any common factors.
Let's analyze the expression (x^2 - 3x - 33) / (x + 4).
Factoring the Numerator
The numerator, x^2 - 3x - 33, is a quadratic expression. We can try to factor it into two binomials.
- Step 1: Find two numbers that multiply to give -33 and add up to -3. The numbers -11 and 3 satisfy these conditions.
- Step 2: Rewrite the expression as (x - 11)(x + 3)
The Simplified Expression
Now, we can rewrite the original expression:
(x^2 - 3x - 33) / (x + 4) = (x - 11)(x + 3) / (x + 4)
Since there are no common factors between the numerator and denominator, this is the simplest form of the expression.
Important Considerations
- Restrictions: It's important to note that the expression is undefined when the denominator is zero. Therefore, x cannot equal -4.
- Long Division: If the numerator is not factorable, or if the factors do not simplify with the denominator, we can use polynomial long division to simplify the expression.
By understanding the process of factoring and simplifying rational expressions, you can effectively manipulate these expressions and solve various mathematical problems involving them.