Nth Degree Polynomial Calculator: Find Polynomial Roots

An nth degree polynomial calculator evaluates a polynomial and can help find its roots—the values of x that make the polynomial equal zero. Enter its coefficients and the degree, then use the polynomial’s equation to calculate a value or solve for roots; the example calculator specification below evaluates a polynomial at a chosen x.

Formula

This example uses a cubic, a third-degree polynomial, in the form f(x) = ax³ + bx² + cx + d. The calculator substitutes the entered value of x and adds the terms to produce f(x).

• a is the leading coefficient, multiplying x³.
• b multiplies x², c multiplies x, and d is the constant term.
• x is the input value. The coefficients and x are ordinary numbers; they have no required physical units unless the problem gives them units.

The general nth-degree polynomial is P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a nonnegative integer and aₙ is not zero. The calculator block is specifically a cubic evaluation example, not a general root solver. To find roots, solve P(x) = 0 using an appropriate algebraic or numerical method.

Worked Example

Using the defaults in the calculator block gives f(x) = x³ − 6x² + 11x − 6, evaluated at x = 2.

1. Substitute x = 2: f(2) = 1(2³) − 6(2²) + 11(2) − 6.
2. Evaluate the powers and products: 8 − 24 + 22 − 6.
3. Add the terms: f(2) = 0.

The displayed result is 0. That also means x = 2 is a root of this polynomial. In fact, it factors as (x − 1)(x − 2)(x − 3), so its other roots are 1 and 3.

What an Nth-Degree Polynomial Calculator Does

“Nth degree” describes the highest exponent of x with a nonzero coefficient. For example, 5x⁴ − 2x + 7 has degree 4, while 3x² + 9 has degree 2. A constant nonzero polynomial has degree 0. The zero polynomial, whose every coefficient is zero, is a special case: its degree is undefined.

People use polynomial calculators for several different tasks, and the phrase does not identify a single universal feature. Depending on the tool, it may evaluate P(x), find real or complex roots, factor an expression, or display a graph. Check what the calculator asks you to enter and what it promises to return. A tool that only evaluates the expression will not automatically solve P(x) = 0.

How to Enter a Polynomial

For a coefficient-based calculator, enter coefficients in descending powers, including zeros for missing terms. For example, 2x⁴ + 3x − 5 has coefficients 2, 0, 0, 3, −5. Those zeros preserve the correct positions: they stand for the absent x³ and x² terms.

• Write the expression in descending powers, from xⁿ through the constant term.
• Include the sign with each coefficient. For x² − 4, the coefficients are 1, 0, −4.
• Enter the leading coefficient even when it is 1; omit terms with coefficient zero only if the tool explicitly supports missing terms.
• For evaluation, provide the requested x-value. For root-finding, make sure the calculator is set to solve the equation P(x) = 0.

If a calculator accepts an equation rather than a coefficient list, enter the polynomial equal to zero when seeking roots. For example, enter x² − 5x + 6 = 0, not simply the number 0 or an unrelated value of x.

Evaluating a Polynomial Versus Finding Its Roots

Evaluation answers “What is P(x) at this particular x?” Root-finding answers “Which x-values make P(x) equal zero?” These are related but different calculations. A value of zero from evaluation confirms a root for that input, but a nonzero result does not tell you how many roots the polynomial has or where they are.

The fundamental theorem of algebra says that a degree-n polynomial with complex coefficients has n complex roots when counted with multiplicity. This does not mean every polynomial has n distinct real roots. For instance, x² + 1 has two complex roots, i and −i, but no real roots; (x − 2)² has a repeated root at 2 rather than two distinct root values.

Choosing a Method and Checking Results

Low-degree polynomials often have convenient exact methods. Linear equations can be rearranged directly, and quadratic equations can be solved by factoring, completing the square, or the quadratic formula. Some higher-degree polynomials factor neatly, but there is no general formula using radicals for every polynomial of degree five or higher. Numerical solvers are therefore commonly used for higher degrees and may return approximations.

When a root is reported as a decimal, treat it as an approximation unless the calculator gives an exact form. Substitute the result into the original polynomial: a root should make P(x) close to zero, allowing for rounding. For a graph, roots are the x-intercepts, but repeated roots can touch the axis without crossing it. A graph’s visible scale can also hide roots that are close together or far from the displayed window.

• Check the coefficient order and signs against the original expression.
• Use parentheses for negative inputs and powers where required; for example, (−2)² = 4.
• Do not round coefficients or intermediate steps unnecessarily, especially when roots are close.
• Check whether the result is real or complex and whether it is exact or approximate.

Frequently Asked Questions

What is an nth degree polynomial?
A polynomial is nth degree when its highest power of the variable with a nonzero coefficient is n. For example, 4x³ − x + 8 is third degree.

How do I find the degree from the coefficients?
Read the coefficients from the highest power downward. The degree is the power attached to the first nonzero coefficient. A zero leading coefficient means the actual degree is lower than the power originally listed.

Can a polynomial calculator find complex roots?
Some root calculators can, but not all. Check whether the tool supports complex numbers; a real-only solver may show only real roots or indicate that no real roots were found.

Why does my calculator show fewer roots than the degree?
It may display only distinct real roots. Roots can repeat, be complex, or be approximated in a way that makes nearby roots difficult to distinguish. Check the tool’s settings and whether it counts multiplicity.

What does a zero result mean when evaluating a polynomial?
If P(x) evaluates to zero at an input x, that input is a root of the polynomial. With rounded coefficients or approximate arithmetic, a displayed zero may instead be a value close to zero.

Can every polynomial be factored by hand?
No. Some factor neatly over integers or rational numbers, while others require numerical methods or more advanced techniques. A calculator’s factorization options may also depend on which number system it supports.

Related

• Highway Radar Detectors: 3 Models Compared
• Regal Tank Volume Calculator: Estimate Capacity
• Scuba Tank Volume Calculator and Gas Capacity

in a | Leading coefficient | - | 1 | 0
in b | Quadratic coefficient | - | -6
in c | Linear coefficient | - | 11
in d | Constant coefficient | - | -6
in x | Value of x | - | 2
out value | Polynomial value | - | a*x^3 + b*x^2 + c*x + d | 0