Quartic equation from roots calculator

Enter distinct or repeated values, and it expands the corresponding factors into standard form, returning coefficients in descending-power order as well as a readable equation. This is useful when a problem gives the zeros but asks for the polynomial, when you need to check a factorization, or when software expects a coefficient array rather than root notation. The calculation is deterministic and runs without an external algebra service.

Turn four roots into linear factors

A number r is a root of a polynomial when substituting r for x makes the polynomial equal zero. The factor theorem connects that statement to a linear factor: a root r corresponds to the factor (x - r). This calculator applies that rule to all four supplied values, forming (x - r1)(x - r2)(x - r3)(x - r4). Negative roots naturally produce addition inside a factor, because subtracting a negative value becomes plus. A zero root contributes the factor x. Repeated values are retained, so roots such as 2, 2, -1, and 0 create a polynomial in which 2 has multiplicity two. The result is monic, meaning its leading coefficient is 1. Four linear factors therefore always produce an x^4 term, even if several lower coefficients cancel. Supply the roots as actual JSON numbers, not expressions or symbolic constants. Exactly four finite real numbers are required, which keeps the operation focused on ordinary numeric quartics and makes validation unambiguous.

How the expansion and coefficients are computed

The calculator expands the factors by deterministic coefficient convolution. It begins with the constant polynomial 1. For each root r, it multiplies the current coefficient list by x - r: every existing coefficient contributes once to the next higher-power position and once, multiplied by -r, to the following position. After four passes, the resulting list is [a, b, c, d, e] for ax^4 + bx^3 + cx^2 + dx + e. Because the constructed polynomial is monic, a is always 1. The remaining values also match the elementary symmetric sums: b is the negative sum of the roots, c is the sum of all pairwise products, d is the negative sum of all triple products, and e is the product of all four roots. The response includes both an ordered coefficients array and named standard-form fields. It also includes factor notation, a formatted expression, and an equation ending in zero, making the same calculation convenient for code, worksheets, and visual checking.

Read the result and verify your quartic

Use the coefficients array when passing the polynomial to numerical software: its order is always x^4, x^3, x^2, x, then the constant term. Use standard_form when named fields are clearer, or copy the equation for a written solution. A quick verification is to substitute each original root into the expanded expression; every substitution should produce zero, apart from the tiny rounding effects that can occur with decimal inputs in binary floating-point arithmetic. You can also compare structural clues. The constant term equals the product of the four roots, while the x^3 coefficient is the negative of their sum. If one root is zero, the constant term must be zero. If roots occur as opposite pairs, odd-power terms may cancel. The endpoint performs no equation solving because the roots are already known; its purpose is the reverse operation, constructing and expanding the unique monic quartic associated with those four roots. API requests are priced at $0.002 per item, with one set of four roots counted as one item.