Rational Expression: 4x^2 - 3x + 17/x^3 - 1 + 2x - 1/x^2 + x + 1 + 6/1 - x
Rational expressions are a fundamental concept in algebra, and they can be quite complex. In this article, we will explore the rational expression 4x^2 - 3x + 17/x^3 - 1 + 2x - 1/x^2 + x + 1 + 6/1 - x.
Simplifying the Expression
To simplify this expression, we need to follow the order of operations (PEMDAS):
- Simplify the numerator:
4x^2 - 3x + 17 - Simplify the denominator:
x^3 - 1 + 2x - 1/x^2 + x + 1 + 6/1 - x
Let's break down the numerator and denominator into smaller parts:
Numerator:
4x^2 - 3xis a quadratic expression+ 17is a constant term
Denominator:
x^3 - 1is a cubic expression+ 2xis a linear term- 1/x^2is a rational expression with a quadratic denominator+ x + 1is a linear expression+ 6/1 - xis a rational expression with a linear denominator
Simplifying the Denominator
To simplify the denominator, we can start by combining like terms:
x^3 - 1 + 2x - 1/x^2 + x + 1 + 6/1 - x
= x^3 + 2x - 1/x^2 + x + 7 - x
= x^3 + x^2 - 1/x^2 + 7
Now, we can simplify the rational expression - 1/x^2:
- 1/x^2 = - (1/x^2)
= -1 /(x^2)
Substituting this back into the denominator, we get:
x^3 + x^2 - 1/(x^2) + 7
Simplifying the Expression
Now that we have simplified the denominator, we can rewrite the original expression:
4x^2 - 3x + 17 / x^3 + x^2 - 1/(x^2) + 7
To simplify this expression, we can try to find a common denominator. However, this expression is still quite complex, and it may not be possible to simplify it further.
Conclusion
In this article, we explored the rational expression 4x^2 - 3x + 17/x^3 - 1 + 2x - 1/x^2 + x + 1 + 6/1 - x. We simplified the numerator and denominator, but the expression remains complex. Rational expressions like this one can be challenging to simplify, but they are an essential part of algebra and mathematics.
