-x%5E2-6x%2B9%2Fx%2B3.jpeg "(x-3) X^2-6x+9/x+3")
Simplifying the Rational Expression (x-3)x^2-6x+9 / (x+3)
This article explores the simplification of the rational expression:
(x-3)x^2-6x+9 / (x+3)
We will simplify this expression by factoring the numerator and denominator, then canceling out common factors.
Factoring the Numerator
The numerator, (x-3)x^2-6x+9, is a quadratic expression. We can factor it using the following steps:
- Factor out the common factor (x-3): (x-3) (x^2 - 2x + 3)
- Factor the remaining quadratic expression: (x-3) (x-1)(x-3)
Therefore, the factored form of the numerator is (x-3)^2 (x-1).
Factoring the Denominator
The denominator, (x+3), is already in its simplest factored form.
Simplifying the Expression
Now, let's rewrite the original expression with the factored forms:
(x-3)^2 (x-1) / (x+3)
We can cancel out the common factor (x-3) from the numerator and denominator, resulting in:
(x-3)(x-1) / (x+3)
This is the simplified form of the original expression.
Important Considerations:
- Restrictions: Remember that the original expression is undefined when the denominator is zero. Therefore, x ≠ -3.
- Domain: The domain of the simplified expression is all real numbers except x = -3.
In conclusion, the simplified form of the expression (x-3)x^2-6x+9 / (x+3) is (x-3)(x-1) / (x+3), with the restriction x ≠ -3.