Multiplying Polynomials Calculator
Free Multiplying polynomials Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs.
Formula
(a_n*x^n + ... + a_0)(b_m*x^m + ... + b_0) = sum of a_i * b_j * x^(i+j)
Each term of the first polynomial is multiplied by every term of the second polynomial. The exponents add together and coefficients multiply. Like terms (same degree) are then combined. The resulting polynomial has degree n + m.
Worked Examples
Example 1: Multiplying a Trinomial by a Binomial
Problem: Multiply (3x^2 + 2x + 1)(x - 4).
Solution: Input polynomial 1 as: 3,2,1 (coefficients of 3x^2 + 2x + 1)\nInput polynomial 2 as: 1,-4 (coefficients of x - 4)\n\n3x^2 * x = 3x^3\n3x^2 * (-4) = -12x^2\n2x * x = 2x^2\n2x * (-4) = -8x\n1 * x = x\n1 * (-4) = -4\n\nCombine like terms:\n3x^3 + (-12 + 2)x^2 + (-8 + 1)x + (-4)\n= 3x^3 - 10x^2 - 7x - 4\n\nVerify at x=2: (12+4+1)(2-4) = 17*(-2) = -34\n3(8) - 10(4) - 7(2) - 4 = 24 - 40 - 14 - 4 = -34. Correct!
Result: (3x^2 + 2x + 1)(x - 4) = 3x^3 - 10x^2 - 7x - 4
Example 2: Multiplying Two Quadratics
Problem: Multiply (x^2 + 3)(x^2 - 2x + 5).
Solution: Input polynomial 1 as: 1,0,3 (x^2 + 0x + 3)\nInput polynomial 2 as: 1,-2,5 (x^2 - 2x + 5)\n\nx^2 * x^2 = x^4\nx^2 * (-2x) = -2x^3\nx^2 * 5 = 5x^2\n3 * x^2 = 3x^2\n3 * (-2x) = -6x\n3 * 5 = 15\n\nCombine: x^4 - 2x^3 + (5+3)x^2 - 6x + 15\n= x^4 - 2x^3 + 8x^2 - 6x + 15\n\nDegree check: 2 + 2 = 4. Correct!
Result: (x^2 + 3)(x^2 - 2x + 5) = x^4 - 2x^3 + 8x^2 - 6x + 15
Frequently Asked Questions
How do you enter polynomials in coefficient form?
Multiplying Polynomials Calculator accepts polynomials as comma-separated coefficients listed from the highest degree term to the lowest (constant term). For example, the polynomial 3x^2 + 2x + 1 is entered as 3,2,1, and 5x^3 - x + 7 is entered as 5,0,-1,7, where the zero represents the missing x^2 term. It is important to include zeros for any missing degree terms because the position of each number determines which power of x it multiplies. The first number is always the leading coefficient (highest power), and the last number is always the constant term (x^0). This format makes it easy to input polynomials of any degree and handles all coefficient values including negatives and decimals.
What is the degree of the product of two polynomials?
The degree of the product polynomial always equals the sum of the degrees of the two input polynomials, provided neither polynomial is the zero polynomial. This follows from the fact that the highest-degree term in the product comes from multiplying the leading terms of each polynomial, and exponents add under multiplication. If you multiply a degree 3 polynomial by a degree 2 polynomial, the result is always degree 5. This rule helps you verify your answer: if (2x^3 + x)(4x^2 - 1) does not give a degree 5 result, something went wrong. This property also determines the number of coefficients in the result, which has (degree + 1) terms at most. Understanding this helps allocate space in array-based polynomial representations.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Can I use Multiplying Polynomials Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.