Polynomial Factoring
Factor polynomials: GCF a(x+y), difference of squares a²−b²=(a+b)(a−b), sum of cubes a³+b³=(a+b)(a²−ab+b²), difference of cubes a³−b³=(a−b)(a²+ab+b²). Grouping and rational root theorem for higher degrees.
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Always factor out GCF first before applying other techniques. a²−b²=(a+b)(a−b) — conjugate pair, middle cancels when expanded. Rational root theorem: possible roots are ±(factors of constant)/(factors of leading coeff).
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Why: Factoring reveals roots, simplifies rational expressions, and solves equations. GCF first, then special patterns. Rational root theorem finds candidate roots for higher-degree polynomials.
How: GCF: factor out common terms. Difference of squares: a²−b²=(a+b)(a−b). Cubes: use sum/difference formulas. Grouping: pair terms, factor each pair, factor common binomial. Rational root: test ±p/q where p|constant, q|leading.
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📐 Examples — Click to Load
Roots
Factor Contribution
📐 Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
x²−4=(x+2)(x−2). Difference of squares with a=x, b=2.
— Example
Rational root theorem: for x³−2x+1, test ±1. P(1)=0 so (x−1) is a factor.
— Theorem
📋 Key Takeaways
- • Factoring hierarchy: GCF first, then special patterns (squares, cubes), then grouping.
- • Difference of squares: a²−b² = (a+b)(a−b). Use when polynomial has form x²−c.
- • Sum/difference of cubes: a³±b³ = (a±b)(a²∓ab+b²).
- • Factor by grouping: Group terms, factor each group, factor out common binomial.
- • Rational root theorem: Rational roots are ±(factor of constant)/(factor of leading coeff).
💡 Did You Know?
📖 Factoring Hierarchy
1. GCF: Factor out the greatest common factor of all terms. 2. Special patterns: Difference of squares, sum/difference of cubes, perfect square trinomials. 3. Grouping: For 4 terms, group and factor. 4. Rational root theorem: For cubics and higher, test ±(factors of constant)/(factors of leading coefficient).
Special Formulas
a²−b² = (a+b)(a−b)
a³+b³ = (a+b)(a²−ab+b²)
a³−b³ = (a−b)(a²+ab+b²)
🎯 Expert Tips
💡 GCF First
Always extract the GCF before applying other methods.
💡 Recognize Patterns
x²−25, x³−8, x⁴−16 are classic patterns.
💡 Verify by Expanding
Multiply factors back to check your answer.
💡 Rational Root Theorem
For cubics+, test ±p/q where p|constant, q|leading.
📊 Factoring Strategies Table
| Pattern | When to Use |
|---|---|
| GCF | Common factor in all terms |
| Difference of squares | x²−a² or similar |
| Sum/diff of cubes | x³±a³ |
| Factor by grouping | 4 terms, pair and factor |
| Rational root theorem | Cubic or higher, find rational roots |
| Nested diff squares | x⁴−a⁴ = (x²+a²)(x²−a²) |
❓ FAQ
What is the factoring hierarchy?
GCF first, then special patterns (squares, cubes), then grouping, then rational root theorem for higher degrees.
When do I use difference of squares?
When the polynomial has the form a²−b², e.g., x²−25 = (x+5)(x−5).
What is the rational root theorem?
If p/q (in lowest terms) is a rational root of P(x), then p divides the constant term and q divides the leading coefficient.
How do I enter coefficients?
Comma-separated, highest degree first. E.g., 6x²+9x is 6,9,0.
What are irreducible polynomials?
Polynomials that cannot be factored into non-constant factors over the rationals. They may have complex roots.
Why factor out GCF first?
It simplifies the polynomial and often reveals patterns like difference of squares that were hidden.
⚠️ Note: Coefficients must be comma-separated, highest degree first. Use 0 for missing terms. Supports polynomials up to degree 4.
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