In algebra, a monomial is a fundamental building block. It is a single term that consists of a constant (a number) multiplied by one or more variables, each raised to a non-negative integer exponent. Examples include 5x^2, -7y, 3a^3b^4, and even just a number like 10 (which can be thought of as 10x^0). Monomials do not contain variables in the denominator or variables raised to fractional or negative powers.
The ability to perform operations on monomials is crucial for simplifying algebraic expressions and solving more complex equations. This calculator handles the four basic arithmetic operations: addition, subtraction, multiplication, and division.
Monomial Operations Explained:
Addition and Subtraction: These operations can only be performed on like monomials. Like monomials have the exact same variables raised to the exact same powers. To add or subtract like monomials, you simply add or subtract their coefficients (the numerical part) and keep the variable part the same.
Example:3x^2y + 5x^2y = (3+5)x^2y = 8x^2y Example:7ab - 2ab = (7-2)ab = 5ab If monomials are not like terms, they cannot be combined further and are simply written next to each other, e.g., 3x^2 + 2y.
Multiplication: Monomials can always be multiplied. To multiply two monomials, you multiply their coefficients and add the exponents of like variables.
Example:(4x^2y) * (3xy^3) = (4*3) * (x^2*x^1) * (y^1*y^3) = 12 * x^(2+1) * y^(1+3) = 12x^3y^4
Division: Monomials can also be divided. To divide two monomials, you divide their coefficients and subtract the exponents of like variables (numerator exponent minus denominator exponent).
Example:(15a^4b^3) / (3a^2b) = (15/3) * (a^4/a^2) * (b^3/b^1) = 5 * a^(4-2) * b^(3-1) = 5a^2b^2 Division by zero (specifically, a monomial that evaluates to zero) is undefined. Also, if the division results in a variable with a negative exponent, it's often rewritten with the variable in the denominator to maintain non-negative exponents.
Use Cases:
Monomial operations are fundamental in:
Simplifying complex algebraic expressions.
Factoring polynomials.
Solving systems of equations.
Understanding polynomial functions and their behavior.
Physics and engineering for representing quantities with units and dimensions.
This calculator provides a quick way to perform these operations, helping students and professionals alike to verify their calculations and better understand algebraic manipulation.
function parseMonomial(monomialStr) {
monomialStr = monomialStr.trim();
if (!monomialStr) {
return { coefficient: NaN, variables: {} };
}
var coefficient = 1;
var variables = {};
var regex = /^([+-]?\d*\.?\d*)\s*([a-zA-Z]\^?\d*)*$/;
var match = monomialStr.match(/^([+-]?\d*\.?\d*)?(.*)$/);
var coeffPart = match[1];
var varPart = match[2];
if (coeffPart === undefined || coeffPart === "" || coeffPart === "+" || coeffPart === "-") {
coefficient = parseFloat(coeffPart + "1");
if (isNaN(coefficient)) {
coefficient = 1;
}
} else {
coefficient = parseFloat(coeffPart);
if (isNaN(coefficient)) {
return { coefficient: NaN, variables: {} }; // Invalid coefficient
}
}
if (varPart) {
var varRegex = /([a-zA-Z])(\^(\d+))?/g;
var varMatch;
while ((varMatch = varRegex.exec(varPart)) !== null) {
var variableName = varMatch[1];
var exponent = varMatch[3] ? parseInt(varMatch[3]) : 1;
if (isNaN(exponent) || exponent < 0) {
return { coefficient: NaN, variables: {} }; // Invalid exponent
}
variables[variableName] = exponent;
}
}
return { coefficient: coefficient, variables: variables };
}
function stringifyMonomial(coefficient, variables) {
if (isNaN(coefficient)) {
return "Invalid";
}
if (Object.keys(variables).length === 0) {
return String(coefficient);
}
var coeffStr = "";
if (coefficient === 1) {
// Don't show 1 if there are variables
} else if (coefficient === -1) {
coeffStr = "-";
} else {
coeffStr = String(coefficient);
}
var varParts = [];
var sortedVars = Object.keys(variables).sort();
for (var i = 0; i < sortedVars.length; i++) {
var variableName = sortedVars[i];
var exponent = variables[variableName];
if (exponent === 1) {
varParts.push(variableName);
} else {
varParts.push(variableName + "^" + exponent);
}
}
return coeffStr + varParts.join('');
}
function areLikeTerms(vars1, vars2) {
var keys1 = Object.keys(vars1);
var keys2 = Object.keys(vars2);
if (keys1.length !== keys2.length) {
return false;
}
for (var i = 0; i 0) {
simplifiedVars[key] = variables[key];
}
}
return { coefficient: coefficient, variables: simplifiedVars };
}
function calculateMonomial() {
var monomial1Str = document.getElementById("monomial1").value;
var monomial2Str = document.getElementById("monomial2").value;
var operation = document.getElementById("operation").value;
var errorMessageDiv = document.getElementById("errorMessage");
var resultDiv = document.getElementById("calculationResult");
errorMessageDiv.textContent = "";
resultDiv.textContent = "–";
var m1 = parseMonomial(monomial1Str);
var m2 = parseMonomial(monomial2Str);
if (isNaN(m1.coefficient) || isNaN(m2.coefficient)) {
errorMessageDiv.textContent = "Invalid monomial format. Please check coefficients and exponents.";
return;
}
var resultCoefficient;
var resultVariables = {};
if (operation === "add" || operation === "subtract") {
if (!areLikeTerms(m1.variables, m2.variables)) {
errorMessageDiv.textContent = "Addition/Subtraction requires like terms.";
// For display, just return the two terms combined
if (operation === "add") {
resultDiv.textContent = stringifyMonomial(m1.coefficient, m1.variables) + " + " + stringifyMonomial(m2.coefficient, m2.variables);
} else { // subtract
resultDiv.textContent = stringifyMonomial(m1.coefficient, m1.variables) + " – " + stringifyMonomial(m2.coefficient, m2.variables);
}
return;
}
resultCoefficient = (operation === "add") ? m1.coefficient + m2.coefficient : m1.coefficient – m2.coefficient;
resultVariables = m1.variables; // Since they are like terms, variables are the same
if (resultCoefficient === 0) {
resultDiv.textContent = "0"; // Special case for zero result
} else {
resultDiv.textContent = stringifyMonomial(resultCoefficient, resultVariables);
}
} else if (operation === "multiply") {
resultCoefficient = m1.coefficient * m2.coefficient;
resultVariables = mergeVariables(m1.variables, m2.variables, 'multiply');
var simplified = simplifyMonomial(resultCoefficient, resultVariables);
resultDiv.textContent = stringifyMonomial(simplified.coefficient, simplified.variables);
} else if (operation === "divide") {
if (m2.coefficient === 0) {
errorMessageDiv.textContent = "Division by zero is undefined.";
return;
}
resultCoefficient = m1.coefficient / m2.coefficient;
resultVariables = mergeVariables(m1.variables, m2.variables, 'divide');
// Check if division resulted in negative exponents, which might require specific formatting
var hasNegativeExponent = false;
for(var key in resultVariables) {
if (resultVariables[key] < 0) {
hasNegativeExponent = true;
break;
}
}
if(hasNegativeExponent) {
errorMessageDiv.textContent = "Division resulted in negative exponents. Consider rewriting.";
// For now, just show the result, potentially with negative exponents if not simplified away
var simplified = simplifyMonomial(resultCoefficient, resultVariables);
resultDiv.textContent = stringifyMonomial(simplified.coefficient, simplified.variables);
} else {
var simplified = simplifyMonomial(resultCoefficient, resultVariables);
resultDiv.textContent = stringifyMonomial(simplified.coefficient, simplified.variables);
}
}
}