Rational Expression Evaluator
A rational expression is a fraction where both the numerator and the denominator are polynomials. This calculator helps you evaluate a rational expression of the form (Ax² + Bx + C) / (Dx² + Ex + F) for a given value of x.
Understanding how to evaluate rational expressions is crucial for determining their value at specific points, identifying domain restrictions (where the denominator is zero), and analyzing their behavior.
How to Use This Calculator:
- Input Numerator Coefficients: Enter the values for A, B, and C corresponding to the
x² term, x term, and constant term in the numerator polynomial, respectively. If a term is missing, enter 0 for its coefficient.
- Input Denominator Coefficients: Similarly, enter the values for D, E, and F for the denominator polynomial.
- Input X Value: Enter the specific numerical value for
x at which you want to evaluate the rational expression.
- Click "Calculate Value": The calculator will compute the value of the expression at the given
x.
Understanding Rational Expressions and Evaluation:
A rational expression is essentially a ratio of two polynomials. For example, (x² + 3x - 4) / (2x + 1) is a rational expression. Evaluating such an expression means substituting a specific number for the variable x and then performing the arithmetic operations to find the numerical result.
Domain Restrictions:
A critical aspect of rational expressions is their domain. The domain of a rational expression includes all real numbers for which the denominator is not equal to zero. If the denominator evaluates to zero for a given x value, the expression is undefined at that point. This calculator will indicate if the expression is undefined due to a zero denominator.
Example Calculation:
Let's evaluate the rational expression (2x² + 3x - 1) / (x² - 4) when x = 3.
- Numerator:
2x² + 3x - 1
- Denominator:
x² - 4
- Value of x:
x = 3
Using the calculator:
- Numerator:
2*(3)² + 3*(3) - 1 = 2*9 + 9 - 1 = 18 + 9 - 1 = 26
- Denominator:
1*(3)² + 0*(3) - 4 = 9 - 4 = 5
- Result:
26 / 5 = 5.2
The calculator would output: "The value of the rational expression is: 5.2"
Example of Undefined Expression:
Consider the expression (x + 1) / (x - 2) when x = 2.
- Numerator:
x + 1 (A=0, B=1, C=1)
- Denominator:
x - 2 (D=0, E=1, F=-2)
- Value of x:
x = 2
Using the calculator:
- Numerator:
0*(2)² + 1*(2) + 1 = 3
- Denominator:
0*(2)² + 1*(2) - 2 = 0
Since the denominator is zero, the expression is undefined at x = 2. The calculator would output: "The rational expression is undefined (denominator is zero)."
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function calculateRationalExpression() {
var numA = parseFloat(document.getElementById('numA').value);
var numB = parseFloat(document.getElementById('numB').value);
var numC = parseFloat(document.getElementById('numC').value);
var denD = parseFloat(document.getElementById('denD').value);
var denE = parseFloat(document.getElementById('denE').value);
var denF = parseFloat(document.getElementById('denF').value);
var xValue = parseFloat(document.getElementById('xValue').value);
var resultDiv = document.getElementById('result');
resultDiv.className = 'calculator-result'; // Reset class for new calculation
// Input validation
if (isNaN(numA) || isNaN(numB) || isNaN(numC) || isNaN(denD) || isNaN(denE) || isNaN(denF) || isNaN(xValue)) {
resultDiv.innerHTML = 'Please enter valid numbers for all fields.';
resultDiv.className += ' error';
return;
}
// Calculate numerator value: Ax^2 + Bx + C
var numeratorValue = numA * Math.pow(xValue, 2) + numB * xValue + numC;
// Calculate denominator value: Dx^2 + Ex + F
var denominatorValue = denD * Math.pow(xValue, 2) + denE * xValue + denF;
if (denominatorValue === 0) {
resultDiv.innerHTML = 'The rational expression is undefined (denominator is zero).';
resultDiv.className += ' error';
} else {
var finalResult = numeratorValue / denominatorValue;
resultDiv.innerHTML = 'The value of the rational expression is:
' + finalResult.toFixed(4) + '';
}
}