Calculator Input
Enter expanded polynomials. Supported examples include x^2-9, 2x^2-8x+6, and 3/2x^3-x+4.
Example Data Table
| Numerator | Denominator | Common Factor | Simplified Form | Restriction |
|---|---|---|---|---|
| x^2 - 9 | x^2 - 3x | x - 3 | (x + 3) / x | x ≠ 0, x ≠ 3 |
| x^2 - 4 | x^2 + x - 6 | x + 2 | (x - 2) / (x - 3) | x ≠ -2, x ≠ 3 |
| 2x^2 + 6x | 4x | 2x | (x + 3) / 2 | x ≠ 0 |
| x^3 - x | x^2 - 1 | x^2 - 1 | x | x ≠ -1, x ≠ 1 |
Formula Used
A rational expression has the form P(x) / Q(x), where Q(x) ≠ 0. The calculator finds a common factor G(x) from the numerator and denominator.
P(x) / Q(x) = [G(x) × A(x)] / [G(x) × B(x)] = A(x) / B(x), where G(x) ≠ 0.
Domain restrictions still come from the original denominator. A canceled factor creates a hole, not permission to use that excluded value.
How to Use This Calculator
- Type the numerator polynomial in expanded form.
- Type the denominator polynomial in expanded form.
- Choose the variable, such as x.
- Set the graph range if needed.
- Press the simplify button.
- Review canceled factors, restrictions, and final form.
- Use CSV or PDF buttons to save your work.
Understanding Rational Expression Simplification
What It Means
A rational expression is a fraction made from polynomials. The numerator is one polynomial. The denominator is another polynomial. Simplifying means reducing the expression without changing its value on the allowed domain. This usually requires finding shared factors. Those factors can be canceled only when they appear as complete factors. Individual terms cannot be canceled safely.
Why Factoring Matters
Factoring reveals hidden structure. For example, x squared minus nine becomes x minus three times x plus three. If the denominator also contains x minus three, that common factor can be removed. The remaining expression is simpler. It is easier to graph, compare, evaluate, and use in later algebra steps.
Restrictions Stay Important
The original denominator controls the domain. Any value that makes the original denominator zero must be excluded. This rule still applies after cancellation. A canceled factor often creates a hole in the graph. The simplified formula may look valid there, but the original expression was undefined.
How This Tool Helps
This calculator parses polynomial input. It computes a greatest common polynomial factor. It divides both parts by that factor. It also reduces shared numeric content. Then it reports the simplified expression and domain restrictions. The graph gives a visual check. The step table explains the process in a clear order.
Best Practice
Use expanded polynomial input for reliable results. Check signs carefully. Keep the same variable in both fields. Review restrictions before using the answer in an equation. When solving an equation, remember that excluded values cannot become final answers. Export your result when you need notes, reports, or homework records.
FAQs
1. What is a rational expression?
A rational expression is a fraction with polynomials in the numerator, denominator, or both. The denominator cannot equal zero.
2. What does simplifying a rational expression mean?
It means canceling common polynomial factors and reducing numeric factors while keeping the same value for allowed inputs.
3. Can I cancel terms across plus signs?
No. You may cancel only complete factors. Terms separated by addition or subtraction must be factored first.
4. Why are domain restrictions shown?
Restrictions show values that make the original denominator zero. These values remain excluded even after a factor is canceled.
5. What input format should I use?
Use expanded polynomial form, such as x^2-9 or 2x^2-8x+6. Fractions like 3/2x are supported.
6. Does the graph show holes?
The graph skips undefined points near denominator zeros. Always read the restrictions to identify holes and vertical exclusions.
7. Can this handle higher degree polynomials?
Yes, it can simplify higher degree polynomial expressions when common factors are found through polynomial division.
8. Why export CSV or PDF?
CSV is useful for spreadsheets. PDF is useful for printing, sharing, or saving a clean summary of the work.