polynomial with roots calculator

What this polynomial with roots calculator does

This tool builds an expanded polynomial when you already know its roots. If your roots are r₁, r₂, ..., rₙ, the polynomial has the factorized form: P(x) = a(x - r₁)(x - r₂)...(x - rₙ), where a is the leading coefficient. The calculator multiplies those factors for you and returns:

• The polynomial degree
• The factored form
• The expanded form
• The coefficient vector [aₙ, aₙ₋₁, ..., a₀]
• A root check table showing that P(r) is approximately zero

How it works behind the scenes

1) Build the monic polynomial from roots

The calculator starts with 1 and repeatedly multiplies by each factor (x - r). This creates a monic polynomial (leading coefficient 1).

2) Apply the leading coefficient

If you enter a leading coefficient a, every coefficient in the monic polynomial is multiplied by a. This scales the graph vertically and sets the leading term to axⁿ.

3) Format and verify

Finally, the calculator formats terms neatly and evaluates the polynomial at each entered root. For decimal roots, tiny floating-point rounding differences are normal, so values extremely close to zero are treated as zero.

Input tips

• Separate roots with commas: 2, -1, 3
• Use fractions if needed: 1/2, -3/4, 5
• Repeat roots for multiplicity: 2, 2, -1 means root 2 has multiplicity 2
• Leading coefficient cannot be zero

Example walkthrough

Example A: roots 2, -1, 3 with leading coefficient 1

The factored form is (x - 2)(x + 1)(x - 3). Expanding gives: x³ - 4x² + x + 6. So the coefficient vector is [1, -4, 1, 6].

Example B: roots 1/2, 1/2, -4 with leading coefficient 8

The repeated root creates a squared factor: 8(x - 1/2)²(x + 4). The output polynomial is cubic with coefficients scaled by 8.

Why this is useful

A polynomial from roots calculator is handy in algebra, precalculus, and numerical methods. It helps when you:

• Need a polynomial model with specific x-intercepts
• Check homework for expanded vs factored forms
• Explore root multiplicity and graph behavior
• Practice Vieta’s formulas by comparing roots and coefficients

Quick FAQ

Does this support complex roots?

This version is designed for real-number roots (including fractions). Complex root support can be added in a future version.

What if I enter decimal roots?

Decimals are fully supported. Coefficients may appear as decimals and can include tiny rounding artifacts due to floating-point arithmetic.

What does multiplicity mean?

If a root appears more than once, that root has multiplicity greater than 1. For example, roots 3, 3, -2 produce factors (x - 3)²(x + 2).

Final note

If you’re studying polynomial equations, this calculator is a fast way to move between roots, factors, and expanded coefficients. Try a few custom root sets and compare how the degree, signs, and intercepts change.