Get Polynomial From Degrees and 0 Calculator

This calculator helps you construct a polynomial equation from given degrees and roots (zeros). Enter the degree of the polynomial and its roots to get the polynomial expression.

What is a Polynomial?

A polynomial is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial is:

P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

Where:

n is the degree of the polynomial
a_n, a_n-1, ..., a₀ are coefficients
x is the variable

Polynomials are fundamental in algebra and have applications in various fields including physics, engineering, and computer science.

How to Find a Polynomial from Degrees and Roots

To construct a polynomial from its degree and roots, you can use the factored form of the polynomial. If you know the roots (x = r₁, x = r₂, ..., x = r_n), the polynomial can be written as:

P(x) = a(x - r₁)(x - r₂) ... (x - r_n)

Where a is the leading coefficient. If the leading coefficient is 1, the polynomial is called monic.

Steps to Find the Polynomial

Identify the degree of the polynomial (n)
List all the roots (r₁, r₂, ..., r_n)
Write the polynomial in factored form: P(x) = a(x - r₁)(x - r₂) ... (x - r_n)
Expand the factored form to get the standard polynomial form

For example, if you have a polynomial of degree 2 with roots at x = 3 and x = -2, the polynomial would be:

P(x) = (x - 3)(x + 2) = x² - x - 6

Example Calculation

Let's find a polynomial of degree 3 with roots at x = 1, x = -2, and x = 3.

P(x) = (x - 1)(x + 2)(x - 3)

First, multiply the first two factors:

(x - 1)(x + 2) = x² + 2x - x - 2 = x² + x - 2

Now multiply the result by the third factor:

(x² + x - 2)(x - 3) = x³ - 3x² + x² - 3x - 2x + 6 = x³ - 2x² - 5x + 6

So the polynomial is:

P(x) = x³ - 2x² - 5x + 6

FAQ

What is the difference between a polynomial and a quadratic equation?: A quadratic equation is a special case of a polynomial with degree 2. Polynomials can have any degree, while quadratic equations specifically have degree 2.
Can a polynomial have complex roots?: Yes, polynomials can have complex roots. The Fundamental Theorem of Algebra states that every non-zero polynomial with complex coefficients has at least one complex root.
How do I find the roots of a polynomial if I have the polynomial expression?: You can use methods like factoring, completing the square, or numerical methods such as the Newton-Raphson method to find the roots of a polynomial.
What is the leading coefficient of a polynomial?: The leading coefficient is the coefficient of the highest degree term in a polynomial. For the polynomial P(x) = a_nxⁿ + ..., the leading coefficient is a_n.