Use ^ for exponents. Multiplication may be implied (e.g., 2x).
Used for “Substitute only” or “Evaluate”. Fractions allowed.
Result
Example Data Table
| # | Input | Suggested Mode | Expected Result |
|---|---|---|---|
| 1 | (x+3)(x-2) | Expand | x^2 + x - 6 |
| 2 | (2x+3)^2 | Expand + Simplify | 4x^2 + 12x + 9 |
| 3 | x^2 - 1 | Factor | (x-1)(x+1) |
| 4 | a^2 + 2ab + b^2 | Factor | (a+b)^2 |
| 5 | (a+b)^5 | Expand | a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5 |
Formulas Used
- Distributive property: a(b+c)=ab+ac; FOIL for two binomials.
- Binomial theorem: (a+b)n = Σ C(n,k)an-kbk.
- Difference of squares: a2−b2=(a−b)(a+b).
- Perfect square trinomials: a2±2ab+b2=(a±b)2.
- Common factoring: Factor out GCF before special patterns.
Symbolic algebra is handled by a client-side CAS to expand, factor, and simplify.
How to Use This Calculator
- Enter an expression like (x+2)^3(x-1) or x^2-4.
- Select the mode: Expand, Factor, Simplify, Substitute, or Evaluate.
- Optionally set substitutions, e.g., x=2, y=3/5.
- Choose precision and toggles for ordering and combining terms.
- Click Calculate to get the result and view steps.
- Use Download CSV to export your history, or Download PDF for a snapshot.
What is Factor Expansion?
Factor expansion converts products or powers of sums into a sum of terms (expansion), or the reverse process—rewriting a polynomial as a product of simpler factors (factorization). The calculator automates both, optionally showing steps.
- Expansion uses distributive rules and the binomial theorem to multiply out brackets.
- Factorization searches for patterns (GCF, squares, cubes) and symbolic factorizations.
- Results may be exact symbolic forms or rounded numeric evaluations, based on options.
When to Expand vs. Factor
| Goal | Choose | Example |
|---|---|---|
| Combine like terms for comparison | Expand | (x+1)(x+2) → x^2+3x+2 |
| Solve polynomial equations | Factor | x^2−5x+6 → (x−2)(x−3) |
| Evaluate efficiently at x=value | Expand | (2x−1)^3 → 8x^3−12x^2+6x−1 |
| Recognize structure or simplify | Factor | a^2+2ab+b^2 → (a+b)^2 |
Tip: Always factor out a greatest common factor before other patterns.
how to calculate starlight travel time factoring universe expansion
Light travel time in an expanding universe is the lookback time from redshift z to today. It requires a cosmological model with parameters like H0, Ωm, ΩΛ, Ωr, and curvature Ωk.
Key formulas (flat ΛCDM shown; include Ωk, Ωr as needed):
H(z) = H0 · √(Ωm(1+z)^3 + Ωr(1+z)^4 + Ωk(1+z)^2 + ΩΛ) Lookback time: t_L(z) = ∫₀ᶻ dz' / ((1+z')·H(z')) Comoving distance: D_c(z) = c ∫₀ᶻ dz' / H(z') Luminosity distance: d_L = (1+z)·D_c Angular diameter distance: d_A = D_c/(1+z)
Practical steps:
- Choose cosmological parameters (e.g., flat ΛCDM: Ωk=0, Ωr≈0).
- Form H(z) and evaluate numerically across 0→z.
- Compute t_L with the integral above; convert to Gyr.
- Optionally compute D_c, d_L, d_A for distance comparisons.
- Interpret: t_L is the starlight’s travel time to us today.
Note: Units matter. Convert H0 from km·s⁻¹·Mpc⁻¹ to s⁻¹ and use c = 299,792,458 m·s⁻¹ consistently.