Understanding Linear Factor Products
A polynomial can be written as a product of linear factors when its zeros are known. Each zero tells where the graph touches or crosses the x-axis. If r is a zero, then x minus r is a factor. This calculator turns that rule into a complete polynomial form.
Why Zeros Matter
Zeros give structure. They show the values that make the polynomial equal zero. A repeated zero creates a repeated factor. Its multiplicity changes the graph shape near that zero. Odd multiplicity usually crosses the axis. Even multiplicity often touches and turns.
Using Multiplicity
Multiplicity is the number of times a factor appears. For example, zero 3 with multiplicity 2 gives the factor (x - 3)^2. Zero -4 with multiplicity 1 gives (x + 4). The calculator accepts several zeros and matching multiplicities. It then builds the factored product.
Leading Coefficient Control
The leading coefficient stretches or flips the polynomial. A positive value keeps the end behavior standard for its degree. A negative value reflects it. You may enter the coefficient directly. You may also use a known point. The tool finds the coefficient that makes the polynomial pass through that point.
Expanded Form
Factored form is useful for roots. Expanded form is useful for standard algebra work. This calculator multiplies every linear factor and lists the expanded polynomial. It also reports degree, leading coefficient, constant term, and optional evaluated value.
Practical Uses
Students can check homework quickly. Teachers can create examples. Writers can build practice problems with controlled roots. The table output helps compare zeros, multiplicities, and factors. The export buttons save results for worksheets, notes, or records.
Accuracy Tips
Use exact fractions when possible. Enter one multiplicity for each zero. Avoid using a known point that is itself a zero when solving for the leading coefficient. That point cannot determine a unique coefficient. Review the step list before copying the final polynomial.
This tool is designed for learning and checking. It does not replace reasoning. It shows the algebra path from zeros to factors, then from factors to a finished polynomial.
With clear inputs, learners can test cases. They can find mistakes and compare formats. They can share final work with accuracy safely online.