Enter Polynomial Coefficients
Use quadratic or linear forms. If a leading coefficient is zero, the expression reduces automatically.
Example Data Table
| Numerator | Denominator | Factored Form | Restrictions | Simplified Form |
|---|---|---|---|---|
| x² - 5x + 6 | x² - x - 6 | [(x - 2)(x - 3)] / [(x - 3)(x + 2)] | x ≠ 3, x ≠ -2 | (x - 2) / (x + 2) |
| x² + x - 6 | x² - 4 | [(x + 3)(x - 2)] / [(x - 2)(x + 2)] | x ≠ 2, x ≠ -2 | (x + 3) / (x + 2) |
| x² - 9 | x² + x - 6 | [(x - 3)(x + 3)] / [(x + 3)(x - 2)] | x ≠ -3, x ≠ 2 | (x - 3) / (x - 2) |
Formula Used
For a rational expression or rational equation set to zero, start with:
R(x) = N(x) / D(x)For quadratic factoring, use the discriminant:
Δ = b² - 4acWhen the discriminant is nonnegative, roots come from:
x = (-b ± √Δ) / (2a)Then write each polynomial in factored form using its roots:
ax² + bx + c = a(x - r₁)(x - r₂)After factoring both numerator and denominator, cancel only common factors:
[(x - p)(x - q)] / [(x - q)(x - s)] = (x - p) / (x - s)Domain restrictions always come from the original denominator, not the simplified one:
D(x) ≠ 0Solutions to the rational equation R(x) = 0 come from the simplified numerator, provided they do not violate original restrictions:
R(x) = 0 ⇒ Simplified Numerator = 0 and Original Denominator ≠ 0Holes occur where a common factor was cancelled. Vertical asymptotes occur where the simplified denominator is zero.
How to Use This Calculator
- Enter the numerator coefficients for ax² + bx + c.
- Enter the denominator coefficients for ax² + bx + c.
- Choose the graph viewing interval with minimum and maximum x-values.
- Click Factor and Analyze.
- Read the factored numerator and denominator forms.
- Check the cancelled factors and the simplified rational form.
- Review restrictions, zeros, holes, asymptotes, and the y-intercept.
- Use the graph to verify sign changes and discontinuities visually.
- Download CSV for graph points or PDF for a shareable report.
Frequently Asked Questions
1. What does this calculator factor?
It factors quadratic or linear polynomials in the numerator and denominator of a rational expression. It then simplifies common factors and analyzes the resulting function or equation.
2. Does cancelling a common factor remove a restriction?
No. A cancelled factor creates a hole, but the original denominator still sets the restriction. That x-value remains excluded from the domain.
3. How are solutions to R(x) = 0 found?
The calculator solves the simplified numerator equal to zero, then removes any x-values that make the original denominator zero. Only valid domain values remain as solutions.
4. What is the difference between a hole and a vertical asymptote?
A hole comes from a cancelled denominator factor. A vertical asymptote comes from a denominator factor that remains after simplification and still makes the function undefined.
5. Can this tool handle irreducible quadratics?
Yes. If a quadratic has no real roots, the calculator marks it as irreducible over the reals. It still evaluates the full rational expression and graph.
6. Why are some graph points missing?
Missing points usually occur near undefined x-values, holes, or vertical asymptotes. The graph intentionally breaks there to avoid false connections across discontinuities.
7. What if the leading quadratic coefficient is zero?
Then the polynomial becomes linear or constant. The calculator automatically reduces the degree and continues the factoring and analysis with the valid expression.
8. What does the CSV export contain?
The CSV file contains sampled x and y values from the plotted rational function. It is useful for graph verification, classroom exercises, and external analysis.