Enter polynomial coefficients to see possible rational zeros.
Understanding the Rational Zero Theorem
The Rational Zero Theorem is a powerful tool in algebra that helps us find possible rational roots (or zeros) of a polynomial with integer coefficients. A rational root is a root that can be expressed as a fraction p/q, where p is an integer and q is a non-zero integer.
For a polynomial of the form:
$a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0$
where all coefficients ($a_n, a_{n-1}, \dots, a_0$) are integers, the Rational Zero Theorem states that any rational root $p/q$ must satisfy two conditions:
p must be a factor of the constant term ($a_0$).
q must be a factor of the leading coefficient ($a_n$).
This theorem doesn't guarantee that any rational roots exist, but it provides a finite list of candidates that we can test (e.g., using synthetic division or by plugging them into the polynomial).
How the Calculator Works:
This calculator takes the coefficients of your polynomial as input.
It identifies the constant term ($a_0$) and the leading coefficient ($a_n$).
It then finds all integer factors of $a_0$ (these are your possible values for 'p').
It finds all integer factors of $a_n$ (these are your possible values for 'q').
Finally, it generates all possible unique fractions $p/q$, including both positive and negative possibilities, which represent the potential rational zeros of the polynomial.
The constant term ($a_0$) is 3. The factors of 3 are: $\pm1, \pm3$. (These are the possible 'p' values).
The leading coefficient ($a_n$) is 2. The factors of 2 are: $\pm1, \pm2$. (These are the possible 'q' values).
The possible rational zeros ($p/q$) are:
From $p=\pm1$: $\pm1/1, \pm1/2 \implies \pm1, \pm0.5$
From $p=\pm3$: $\pm3/1, \pm3/2 \implies \pm3, \pm1.5$
So, the complete list of possible rational zeros for $2x^3 + 3x^2 – 8x + 3$ is: $\pm1, \pm0.5, \pm3, \pm1.5$. The calculator will list these out for you.
function findRationalZeros() {
var coefficientsInput = document.getElementById("coefficients").value;
var resultDiv = document.getElementById("rationalZeros");
if (!coefficientsInput) {
resultDiv.textContent = "Please enter the polynomial coefficients.";
return;
}
var coeffsStr = coefficientsInput.split(/\s+/); // Split by one or more spaces
var coeffs = [];
for (var i = 0; i < coeffsStr.length; i++) {
var num = parseFloat(coeffsStr[i]);
if (isNaN(num)) {
resultDiv.textContent = "Invalid input. Please enter numbers separated by spaces.";
return;
}
coeffs.push(num);
}
// Ensure all coefficients are integers for the theorem to apply directly
for(var i = 0; i < coeffs.length; i++) {
if (!Number.isInteger(coeffs[i])) {
resultDiv.textContent = "The Rational Zero Theorem applies to polynomials with INTEGER coefficients. Please ensure all inputs are integers.";
return;
}
}
if (coeffs.length < 2) {
resultDiv.textContent = "A polynomial must have at least a leading coefficient and a constant term.";
return;
}
var constantTerm = coeffs[coeffs.length – 1];
var leadingCoefficient = coeffs[0];
if (leadingCoefficient === 0) {
resultDiv.textContent = "The leading coefficient cannot be zero.";
return;
}
var pFactors = getFactors(Math.abs(constantTerm));
var qFactors = getFactors(Math.abs(leadingCoefficient));
var possibleZeros = [];
var zeroSet = new Set(); // To store unique values
for (var i = 0; i < pFactors.length; i++) {
for (var j = 0; j < qFactors.length; j++) {
var p = pFactors[i];
var q = qFactors[j];
// Positive p/q
var zeroPos = p / q;
if (!zeroSet.has(zeroPos)) {
possibleZeros.push(zeroPos);
zeroSet.add(zeroPos);
}
// Negative p/q
var zeroNeg = -p / q;
if (!zeroSet.has(zeroNeg)) {
possibleZeros.push(zeroNeg);
zeroSet.add(zeroNeg);
}
}
}
// Sort the results for better readability
possibleZeros.sort(function(a, b) {
return a – b;
});
if (possibleZeros.length === 0) {
resultDiv.textContent = "No possible rational zeros found (constant term or leading coefficient might be zero or other invalid states).";
} else {
resultDiv.textContent = possibleZeros.join(', ');
}
}
function getFactors(n) {
var factors = [1]; // 1 is always a factor
if (n !== 0) {
factors.push(n); // n is also a factor
}
for (var i = 2; i * i <= n; i++) {
if (n % i === 0) {
factors.push(i);
if (i * i !== n) { // Avoid adding the square root twice
factors.push(n / i);
}
}
}
// Add negative factors
var negativeFactors = factors.map(function(f) { return -f; });
return factors.concat(negativeFactors);
}