Consider the following expressions: 1) \(x + {x^2} - \frac{1}{x}\) 2) \(\sqrt {a{x^2} + bx + x - c + \frac{d}{x} - \frac{e}{{{x^2}}}} \) 3) 3x 2 - 5x + ab 4) \(\frac{2}{{{x^2} - ax + {b^3}}}\) 5) \(\frac{1}{x} - \frac{2}{{x + 5}}\) Which of the above are rational expressions?
1, 3, 4 and 5 only
Understanding Rational Expressions
A rational expression is defined as a fraction where both the numerator and the denominator are polynomials. A polynomial is an expression consisting of variables and coefficients that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Constant values are also considered polynomials (specifically, degree 0 polynomials).
In mathematical terms, a rational expression is of the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not the zero polynomial.
Let's analyze each given expression to determine if it fits this definition:
Analyzing Expression 1: \(x + {x^2} - \frac{1}{x}\)
The expression is \(x + x^2 - \frac{1}{x}\). We can combine these terms by finding a common denominator, which is \(x\):
\(x + x^2 - \frac{1}{x} = \frac{x \cdot x}{x} + \frac{x^2 \cdot x}{x} - \frac{1}{x} = \frac{x^2}{x} + \frac{x^3}{x} - \frac{1}{x} = \frac{x^3 + x^2 - 1}{x}\)
Here, the numerator is \(x^3 + x^2 - 1\), which is a polynomial. The denominator is \(x\), which is also a polynomial (of degree 1). Since this expression can be written as a ratio of two polynomials, it is a rational expression.
Analyzing Expression 2: \(\sqrt {a{x^2} + bx + x - c + \frac{d}{x} - \frac{e}{{{x^2}}}} \)
This expression involves a square root around a term that includes variables. For an expression to be rational, the variable cannot be inside a radical (like a square root or cube root) or have fractional exponents (which are equivalent to radicals). Since this expression contains \(\sqrt{...}\) with \(x\) inside, it is not a rational expression.
Analyzing Expression 3: \(3x^2 - 5x + ab\)
The expression is \(3x^2 - 5x + ab\). Assuming \(a\) and \(b\) are constants, this is a polynomial in terms of \(x\). It consists of terms with non-negative integer exponents (\(x^2\), \(x^1\), and \(x^0\)). A polynomial is a special case of a rational expression because it can be written as the polynomial divided by 1 (which is a constant polynomial). For example, \(3x^2 - 5x + ab = \frac{3x^2 - 5x + ab}{1}\).
Therefore, this expression is a rational expression.
Analyzing Expression 4: \(\frac{2}{{{x^2} - ax + {b^3}}}\)
The expression is \(\frac{2}{x^2 - ax + b^3}\). The numerator is 2, which is a constant polynomial. The denominator is \(x^2 - ax + b^3\). Assuming \(a\) and \(b\) are constants, this is a polynomial in terms of \(x\). Since this expression is a ratio of two polynomials, it is a rational expression.
Analyzing Expression 5: \(\frac{1}{x} - \frac{2}{{x + 5}}\)
The expression is \(\frac{1}{x} - \frac{2}{x + 5}\). We can combine these terms by finding a common denominator, which is \(x(x+5)\):
\(\frac{1}{x} - \frac{2}{x + 5} = \frac{1 \cdot (x + 5)}{x \cdot (x + 5)} - \frac{2 \cdot x}{(x + 5) \cdot x} = \frac{x + 5}{x(x + 5)} - \frac{2x}{x(x + 5)} = \frac{(x + 5) - 2x}{x(x + 5)} = \frac{x + 5 - 2x}{x^2 + 5x} = \frac{5 - x}{x^2 + 5x}\)
Here, the numerator is \(5 - x\) (or \(-x + 5\)), which is a polynomial. The denominator is \(x^2 + 5x\), which is also a polynomial. Since this expression can be written as a ratio of two polynomials, it is a rational expression.
Summary of Rational Expressions
Based on our analysis:
- Expression 1 (\(x + x^2 - \frac{1}{x}\)) is a rational expression.
- Expression 2 (\(\sqrt {a{x^2} + ...} \)) is not a rational expression due to the square root containing a variable.
- Expression 3 (\(3x^2 - 5x + ab\)) is a rational expression (it's a polynomial).
- Expression 4 (\(\frac{2}{x^2 - ax + b^3}\)) is a rational expression.
- Expression 5 (\(\frac{1}{x} - \frac{2}{x + 5}\)) is a rational expression.
Therefore, expressions 1, 3, 4, and 5 are rational expressions.
Rational Expression Identification Analysis
| Expression | Form | Is it Rational? | Reason |
|---|---|---|---|
| 1) \(x + x^2 - \frac{1}{x}\) | Can be written as \(\frac{x^3 + x^2 - 1}{x}\) | Yes | Ratio of two polynomials |
| 2) \(\sqrt{...}\) | Involves \(\sqrt{\text{variable}}\) | No | Contains variable under square root |
| 3) \(3x^2 - 5x + ab\) | Polynomial | Yes | Polynomials are rational expressions (\(\frac{P(x)}{1}\)) |
| 4) \(\frac{2}{x^2 - ax + b^3}\) | Ratio of constant polynomial (2) and polynomial (\(x^2 - ax + b^3\)) | Yes | Ratio of two polynomials |
| 5) \(\frac{1}{x} - \frac{2}{x + 5}\) | Can be written as \(\frac{5 - x}{x^2 + 5x}\) | Yes | Ratio of two polynomials |
Revision Table: Rational Expressions
| Concept | Definition/Description | Example |
|---|---|---|
| Polynomial | Sum of terms, each of which is a constant multiplied by one or more variables raised to non-negative integer powers. | \(3x^2 - 5x + 1\), \(y^3 + 2y\), \(5\) |
| Rational Expression | A ratio (fraction) \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) \neq 0\). Polynomials themselves are rational expressions. | \(\frac{x+1}{x-2}\), \(x^2 + 3x\), \(\frac{5}{x}\) |
| Not a Rational Expression | Expressions involving variables under radicals, fractional exponents, or variables in the exponent. | \(\sqrt{x}\), \(x^{1/2}\), \(2^x\), \(\frac{|x|}{x+1}\) |
Additional Information: Domain of Rational Expressions
An important aspect of rational expressions is their domain. The domain of a rational expression \(\frac{P(x)}{Q(x)}\) includes all real numbers for which the denominator \(Q(x)\) is not equal to zero. This is because division by zero is undefined.
For example:
- The domain of \(\frac{1}{x}\) is all real numbers except \(x = 0\).
- The domain of \(\frac{x+5}{x-2}\) is all real numbers except \(x = 2\).
- The domain of \(\frac{x^2 + 1}{x^2 - 4}\) is all real numbers except \(x^2 - 4 = 0\), which means \(x \neq 2\) and \(x \neq -2\).
Understanding the domain is crucial when working with rational expressions, especially when simplifying them or solving equations involving them.
Important Questions from Quadratic Equations
If k = c, then the roots of the equation are:
If \(\rm {k}=\frac{{c}}{2},({c} \neq 0)\), then the roots of the equation are :
What is the number of real roots of the equation?
What is the sum of all the roots of the equation?
If α and β are the distinct roots of equation x2 - x + 1 = 0, then what is the value of \(\left|\frac{\alpha^{100}+\beta^{100}}{\alpha^{100}-\beta^{100}}\right|\) ?
