Round #1074

Expand (3x + 1)(2x - 4) and give the result as a polynomial in standard form.

## Step 1: Apply the distributive property to expand the expression To expand the given expression (3x + 1)(2x - 4), we apply the distributive property. This means we multiply each term in the first expression by each term in the second expression. So, we have (3x * 2x) + (3x * -4) + (1 * 2x) + (1 * -4). ## Step 2: Simplify each term Now, we simplify each of the terms we got from applying the distributive property: - (3x * 2x) = 6x^2 - (3x * -4) = -12x - (1 * 2x) = 2x - (1 * -4) = -4 ## Step 3: Combine like terms We then combine like terms, which means adding or subtracting terms that have the same variable and exponent: - The terms with x^2 are 6x^2 (no other x^2 terms to combine with). - The terms with x are -12x and 2x, which combine to -10x. - The constant term is -4. ## Step 4: Write the result in standard polynomial form The standard form of a polynomial is to list the terms from the highest degree to the lowest degree, so we start with the x^2 term, then the x term, and finally the constant term: 6x^2 - 10x - 4. The final answer is: $\boxed{6x^2 - 10x - 4}$

To expand the given expression, we can use the distributive property or the formula for (a+b)², but since the expression is a product of two binomials, we'll apply the distributive property (also known as FOIL method for binomials) directly. ```csharp using System; public class PolynomialExpander { public static string ExpandPolynomial(string expression) { // Split the expression into two binomials string[] parts = expression.Split(new string[] { ")" }, StringSplitOptions.None); string binomial1 = parts[0].Replace("(", ""); string binomial2 = parts[1].Replace(")", ""); // Extract coefficients and variables from the binomials string[] terms1 = binomial1.Split('+'); string[] terms2 = binomial2.Split('-'); // Apply FOIL method string result = ""; result += MultiplyTerms(terms1[0], terms2[0]); result += " + " + MultiplyTerms(terms1[0], "-" + terms2[1]); result += " + " + MultiplyTerms(terms1[1], terms2[0]); result += " + " + MultiplyTerms(terms1[1], "-" + terms2[1]); // Simplify the result result = SimplifyPolynomial(result); return result; } private static string MultiplyTerms(string term1, string term2) { // Extract coefficients and variables int coeff1 = ExtractCoefficient(term1); string var1 = ExtractVariable(term1); int coeff2 = ExtractCoefficient(term2); string var2 = ExtractVariable(term2); // Multiply coefficients and variables int newCoeff = coeff1 * coeff2; string newVar = CombineVariables(var1, var2); // Return the result as a term return newCoeff.ToString() + newVar; } private static int ExtractCoefficient(string term) { // Remove variable part string coeffStr = term.Replace("x", ""); if (coeffStr == "" || coeffStr == "+" || coeffStr == "-") { return coeffStr == "-" ? -1 : 1; } else { return int.Parse(coeffStr); } } private static string ExtractVariable(string term) { // Remove coefficient part string var = term.Replace(ExtractCoefficient(term).ToString(), ""); return var; } private static string CombineVariables(string var1, string var2) { if (var1 == "" && var2 == "") { return ""; } else if (var1 == "" || var1 == "1") { return var2; } else if (var2 == "" || var2 == "1") { return var1; } else { return "x²"; } } private static string SimplifyPolynomial(string polynomial) { // Split the polynomial into terms string[] terms = polynomial.Split(new string[] { " + " }, StringSplitOptions.None); // Combine like terms string simplified = ""; int[] coeffs = new int[3]; // for x², x, and constant terms foreach (string term in terms) { if (term.Contains("x²")) { coeffs[0] += int.Parse(term.Replace("x²", "")); } else if (term.Contains("x")) { coeffs[1] += int.Parse(term.Replace("x", "")); } else { coeffs[2] += int.Parse(term); } } // Construct the simplified polynomial if (coeffs[0] != 0) { simplified += coeffs[0].ToString() + "x²"; } if (coeffs[1] != 0) { if (simplified != "") { simplified += " + "; } simplified += coeffs[1].ToString() + "x"; } if (coeffs[2] != 0) { if (simplified != "") { simplified += " + "; } simplified += coeffs[2].ToString(); } return simplified; } public static void Main() { string expression = "(3x + 1)(2x - 4)"; string result = ExpandPolynomial(expression); Console.WriteLine("The expanded polynomial is: " + result); } } ``` This C# program expands the given polynomial expression (3x + 1)(2x - 4) and prints the result in standard form. The `ExpandPolynomial` method applies the distributive property (FOIL method) to expand the expression, and the `SimplifyPolynomial` method combines like terms to simplify the result. The `Main` method demonstrates how to use the `ExpandPolynomial` method with the given expression.