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How to enter a polynomial

1. Write the coefficients in order. From the highest degree to the lowest, separated by spaces. E.g. x³−6x²+11x−6 → 1 -6 11 -6.
2. Mark missing terms with a zero. For example, x²−2 is written 1 0 -2.
3. Calculate the roots. You get the roots, the degree and the derivative. Complex roots are shown in the form a ± bi.

The roots and degree of a polynomial

The degree of a polynomial is its highest power, and it tells the maximum number of roots. A second-degree polynomial has at most two roots, a third-degree polynomial three, and so on. Some of the roots may be complex, in which case they occur as conjugate pairs. The calculator uses an established numerical method that finds all the roots at once.

Derivative and use

The calculator also shows the polynomial's derivative, which is obtained with the power rule: the term a·xⁿ becomes n·a·xⁿ⁻¹. The derivative describes the slope of the curve and helps find extrema. Polynomials are used for modelling in engineering, economics and the natural sciences, and their roots correspond to the points where the modelled quantity takes the value zero.

Example: a third-degree polynomial

Consider the polynomial x³ − 6x² + 11x − 6, whose coefficients are entered as 1 -6 11 -6. The calculator finds the roots 1, 2 and 3. You can check one root by substitution: 1³ − 6·1² + 11·1 − 6 = 1 − 6 + 11 − 6 = 0, so x = 1 is a root. The polynomial can then be written in product form (x−1)(x−2)(x−3). A third-degree polynomial has at most three roots, and here they are all real and distinct.

From coefficients to roots

There is a handy connection between the roots and the coefficients. For a second-degree polynomial x² + bx + c, the sum of the roots is −b and the product is c. In the third-degree example above, the sum of the roots 1 + 2 + 3 = 6 corresponds to the coefficient of the second-highest term (with the sign changed), and the product of the roots 1·2·3 = 6 corresponds to the constant term. These relationships help you guess roots and quickly check the calculator's result.

Factoring and polynomial division

The roots and factors of a polynomial are closely connected. If a number r is a root of the polynomial, then (x − r) is a factor of it. For example, when the roots of x² − 5x + 6 are 2 and 3, the polynomial factors into (x − 2)(x − 3). Factoring is an effective way to solve and simplify: from the product form the roots are seen directly, and expressions can be reduced.

Polynomials can also be divided by each other like integers, giving a quotient and a possible remainder. If the division comes out even, the divisor is a factor. This is used, for example, when one root is known and you want to reduce the degree of the polynomial to find the remaining roots. The polynomial calculator gives the roots directly, but understanding the idea of factoring helps you grasp why there is a certain number of roots and how they relate to the coefficients.

Good to know about polynomials

• A degree-n polynomial has exactly n roots when complex and multiple ones are counted.
• The complex roots of a polynomial with real coefficients always occur as conjugate pairs a ± bi.
• The constant term is the value of the polynomial at x = 0.
• If the sum of the coefficients is zero, x = 1 is a root — a handy quick check.

خلاصة

The polynomial calculator finds the roots — including complex ones — as well as the degree and derivative from the coefficients alone. Enter the coefficients from the highest degree to the lowest and mark missing terms with a zero. The tool shows all the roots at once, and with the relationships between roots and coefficients you can quickly check the result.

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