Polynomial From Roots Calculator

Calculator

Roots

Separate roots with commas, semicolons, or new lines. Example: 3, -2, 1, 2+3i

Leading Coefficient

Variable

Decimal Precision

Evaluate At

Graph Points

Graph Minimum x

Graph Maximum x

Actions

Example Data Table

Roots	Leading Coefficient	Factored Form	Expanded Polynomial
3, -2, 1	1	(x - 3)(x + 2)(x - 1)	x^3 - 2x^2 - 5x + 6
2, 2, -4	3	3(x - 2)(x - 2)(x + 4)	3x^3 - 36x + 48
1+i, 1-i	1	(x - (1+i))(x - (1-i))	x^2 - 2x + 2

Formula Used

General formula:

P(x) = a(x - r1)(x - r2)(x - r3)...(x - rn)

a is the leading coefficient.

r1, r2, ... rn are the roots of the polynomial.

The calculator starts with the leading coefficient. It then multiplies each factor by synthetic expansion. The final coefficient list is displayed from the highest power to the constant term.

How to Use This Calculator

Enter all roots in the roots box. Use commas, semicolons, or new lines.
Repeat a root if it has multiplicity greater than one.
Enter the leading coefficient. Use 1 for a monic polynomial.
Choose the variable, precision, graph range, and evaluation value.
Press Calculate Polynomial to see results above the form.
Use CSV or PDF export to save your final results.

Polynomial From Roots Guide

Why This Method Matters

A polynomial from roots calculator turns known zeros into a complete expanded equation. It saves time when algebraic products become long. It also lowers mistakes caused by signs, fractions, and repeated roots.

How Roots Become Factors

Each root describes a factor. If 3 is a root, then x minus 3 is a factor. If -2 is a root, then x plus 2 is a factor. The calculator multiplies these factors in order. It also applies the leading coefficient at the start.

Where It Helps

This tool is useful in algebra, precalculus, calculus, and engineering courses. It helps when building test equations. It also helps when checking graph behavior. Roots show where the curve crosses or touches the x axis. Repeated roots can make the graph bounce, flatten, or turn near the axis.

Complex Roots

Complex roots are also supported. Enter a value like 2+3i or 2-3i. For a polynomial with real coefficients, complex roots usually appear in conjugate pairs. The calculator can expand them and show the resulting coefficients. When conjugate pairs are balanced, imaginary parts cancel.

Reading Results

The coefficient table gives a clear view of every term. It lists powers from highest to lowest. This makes copying the equation easier. It also helps users compare the expanded form with the factor form.

Using the Graph

The graph is a helpful visual check. Real roots should appear near x intercepts. A steep graph may need a wider range. A flat graph may need a smaller range. Change the graph limits to inspect the curve.

Evaluation and Export

The optional evaluation field checks the polynomial at one x value. This is useful for testing a point. It can also verify a root. If the selected value is a root, the result should be close to zero.

Use the CSV export for spreadsheets. Use the PDF export for notes and reports. Both options keep the main results easy to share. Always review rounded values when exact answers matter.

For exact classroom work, keep roots simple. Fractions and radicals may need symbolic handling. This page focuses on dependable numeric expansion. It still shows each computed coefficient with chosen precision. That balance makes it practical for daily study and fast verification. Save the final expression after checking every input carefully.

FAQs

1. What does a polynomial from roots calculator do?

It builds an expanded polynomial from roots and a leading coefficient. It multiplies all factors and shows coefficients, degree, graph data, and exportable results.

2. How do I enter repeated roots?

Enter the same root more than once. For example, use 2, 2, -1 when 2 has multiplicity two and -1 has multiplicity one.

3. Can I use complex roots?

Yes. Enter complex roots like 2+3i, 2-3i, i, or -i. Use conjugate pairs when you need real polynomial coefficients.

4. What is the leading coefficient?

The leading coefficient is the multiplier before all factors. In a(x-r1)(x-r2), the value a controls the leading term and vertical scale.

5. Why does the graph not cross at every root?

A repeated root may make the curve touch and turn instead of crossing. Even multiplicity often bounces. Odd multiplicity often crosses.

6. What does the coefficient table show?

It lists each power, coefficient, and term. This helps you copy the expanded polynomial and check every part of the calculation.

7. Can I download the results?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report containing the main calculated results.

8. Are rounded answers exact?

Rounded answers are practical approximations. Increase decimal precision for more detail. Use symbolic algebra when exact fractional or radical forms are required.