Polynomial Graphing Calculator

A polynomial of degree n has exactly n roots in the complex number system (Fundamental Theorem of Algebra). Some roots may be repeated, and some may be complex (non-real) numbers.

Turning points are where the polynomial changes from increasing to decreasing (or vice versa). A degree-n polynomial has at most n-1 turning points. At turning points, the derivative P'(x) = 0.

Use the Rational Root Theorem to test possible rational roots (factors of a₀ divided by factors of aₙ). Then use synthetic division or polynomial long division to reduce the polynomial and find remaining roots.

End behavior describes what P(x) approaches as x → +∞ and x → -∞. It is determined by the leading term aₙxⁿ. If aₙ > 0 and n is even, both ends go up. If n is odd, the ends go in opposite directions.

Complex roots involve imaginary numbers (i = √(-1)). They always come in conjugate pairs for polynomials with real coefficients. For example, x² + 1 = 0 has roots i and -i.

Yes. For degree 1 (linear), it is a straight line. Degree 2 is a parabola. Degree 3+ have more complex curves. This calculator handles degrees 1 through 5.

Set x = 0 in the polynomial. P(0) = a₀ (the constant term). This is where the graph crosses the y-axis.

If a polynomial has integer coefficients, any rational root p/q must have p as a factor of the constant term and q as a factor of the leading coefficient. This narrows down candidates for testing.

Every non-zero polynomial of degree n has exactly n roots (counting multiplicity) in the complex number system. This guarantees that all polynomials can be factored completely.

After finding all real roots, use polynomial division to reduce the polynomial. The remaining quadratic can be solved using the quadratic formula, which may yield complex roots.

Multiplicity is how many times a root appears as a factor. A root of multiplicity 1 crosses the axis, while a root of even multiplicity touches and turns around.

The leading coefficient determines the vertical stretch and end behavior direction. A positive leading coefficient means both ends go in the same direction (even degree) or opposite (odd degree).

The x-intercepts are the real roots of the polynomial, where the graph crosses the x-axis. Not all polynomials have real x-intercepts (e.g., x² + 1).

The degree is the highest power of x in the polynomial. For example, 3x⁴ + 2x² - 5 has degree 4.

A polynomial function is a function of the form P(x) = aₙxⁿ + ... + a₁x + a₀, where n is a non-negative integer and the coefficients are constants.

Start by looking for common factors. Use the Rational Root Theorem to find linear factors. For remaining factors, use grouping, special formulas (difference of squares), or the quadratic formula.

Synthetic division is a shorthand method for dividing a polynomial by a linear factor (x - c). It is faster than long division and helps find polynomial coefficients after factoring.

If r is a root of P(x), then (x - r) is a factor of P(x). This means P(x) can be written as (x - r) × Q(x), where Q(x) is a polynomial of degree n-1.

Choose a test point in the interval and evaluate P(x). The sign of P at that point determines the sign on the entire interval, since the polynomial can only change sign at its roots.

Real roots are numbers on the real number line where P(x) = 0. Complex roots involve imaginary numbers and are not visible on a real graph. They always appear in conjugate pairs.

Find the derivative P'(x), set it equal to zero, and solve. The solutions give the x-coordinates of potential turning points. Evaluate P(x) at these points to find the y-coordinates.

Local maxima and minima are turning points where the function changes direction. They can be found by analyzing the derivative and second derivative of the polynomial.

Between consecutive real roots, the polynomial is either entirely positive or entirely negative. Test a point in each interval to determine the sign.

A repeated root (or multiple root) occurs when (x - r)² or higher is a factor. The graph touches the x-axis at a repeated root but does not cross it.

The derivative of P(x) = aₙxⁿ + ... + a₁x + a₀ is P'(x) = naₙxⁿ⁻¹ + ... + a₁. It gives the slope of the tangent line at any point.

The Polynomial Graphing Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Polynomial Graphing Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.