Polynomial Multiplication
Multiply polynomials using the distributive property: each term of P₁ times each term of P₂. Degree of product = deg(P₁) + deg(P₂). Leading coefficient = LC(P₁) × LC(P₂). Verify at x=1: P(1) = sum of coefficients.
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Degree of product always equals deg(P₁) + deg(P₂) when both have nonzero leading coefficients. Verification at x=1: P(1) = sum of coefficients — a quick sanity check. Same idea as FOIL but for polynomials of any length — each term × each term.
Ready to run the numbers?
Why: Polynomial multiplication extends FOIL to any degree. Used in signal processing, control theory, and polynomial interpolation. Verification at x=1 checks arithmetic quickly.
How: For each term aᵢxⁱ in P₁ and bⱼxʲ in P₂, add aᵢbⱼxⁱ⁺ʲ to the product. Combine like terms. Degree of product = sum of degrees. At x=1, P(1) = sum of all coefficients.
Run the calculator when you are ready.
📐 Examples — Click to Load
Polynomial 1: a₃x³ + a₂x² + a₁x + a₀
Polynomial 2: b₃x³ + b₂x² + b₁x + b₀
Coefficients
Degree Contribution
📐 Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Degree of product = sum of degrees. (x²+1)(x³+2) has degree 5.
— Algebra
P(1) = sum of coefficients. Quick verification for any polynomial.
— Verification
📋 Key Takeaways
- • Distributive property: Multiply each term in P₁ by each term in P₂
- • Degree of product: deg(P₁×P₂) = deg(P₁) + deg(P₂)
- • Leading coefficient: product of the leading coefficients of P₁ and P₂
- • Like terms: Combine terms with the same power of x
- • Verification: Evaluate P₁(1)×P₂(1) = (P₁×P₂)(1) to check
💡 Did You Know?
📖 How It Works
To multiply P₁(x) × P₂(x), multiply every term in P₁ by every term in P₂. Each product has the form (a·x^i)(b·x^j) = (a·b)·x^(i+j). Then combine like terms (same exponent).
Example: (x+2)(x²+3x+1)
x·x² + x·3x + x·1 + 2·x² + 2·3x + 2·1 = x³ + 3x² + x + 2x² + 6x + 2 = x³ + 5x² + 7x + 2
Degree rule
deg(P₁) + deg(P₂) = deg(product). (x²)(x³) = x⁵. Leading coefficient = product of leading coefficients.
Verification at x=1
P(1) = sum of all coefficients. So (P₁×P₂)(1) = P₁(1)·P₂(1). Quick check!
🎯 Expert Tips
💡 Organize by degree
Write partial products in columns by power of x, then add each column.
💡 Leading coefficient
Always = (leading of P₁) × (leading of P₂).
💡 Verify at x=1
P(1) = sum of coefficients. Cross-check product.
💡 Long multiplication
Stack polynomials, multiply each term, shift, add — like integer multiplication.
📊 Polynomial Multiplication Table
| Rule | Formula | Example |
|---|---|---|
| Distributive | P₁·P₂ = Σ (each term of P₁ × each of P₂) | (x+2)(x+3) = x²+3x+2x+6 |
| Degree | deg(P₁×P₂) = deg(P₁)+deg(P₂) | deg(x²·x³) = 5 |
| Leading coef | LC(product) = LC(P₁)×LC(P₂) | 2x² × 3x = 6x³ |
| Exponent add | x^m · x^n = x^(m+n) | x² · x³ = x⁵ |
❓ FAQ
How do I multiply polynomials?
Multiply each term in the first polynomial by each term in the second. Combine like terms (same power of x).
What is the degree of the product?
deg(P₁×P₂) = deg(P₁) + deg(P₂). Example: (x²)(x³) has degree 5.
What is the leading coefficient?
The coefficient of the highest power. It equals (leading of P₁) × (leading of P₂).
How do I verify my answer?
Evaluate both sides at x=1. P(1) = sum of coefficients. (P₁×P₂)(1) should equal P₁(1)×P₂(1).
What is polynomial long multiplication?
Like integer long multiplication: stack polynomials, multiply each term, shift by degree, add rows.
How does this relate to binomials?
Binomials are degree-1 polynomials. FOIL is a special case of polynomial multiplication.
📝 Quick Reference
📋 Worked Examples
Example 1: (x+2)(x²+3x+1)
x·x² + x·3x + x·1 + 2·x² + 2·3x + 2·1 = x³ + 3x² + x + 2x² + 6x + 2
= x³ + 5x² + 7x + 2
Example 2: (2x²+x)(3x-4)
2x²·3x + 2x²·(-4) + x·3x + x·(-4) = 6x³ − 8x² + 3x² − 4x
= 6x³ − 5x² − 4x
Example 3: (x²+x+1)(x²-x+1)
Conjugate pattern: x²·x² − x²·x + x²·1 + x·x² − x·x + x·1 + 1·x² − 1·x + 1·1
x⁴ − x³ + x² + x³ − x² + x + x² − x + 1. Cancel: x³, x terms.
= x⁴ + x² + 1
📐 Polynomial Long Multiplication
Like integer long multiplication, stack polynomials by degree and multiply each term. Example: (x²+2x+1)(x+3):
x² + 2x + 1
× x + 3
─────────────
3x² + 6x + 3 (×3)
x³ + 2x² + x (×x)
─────────────
x³ + 5x² + 7x + 3
Each row is one term of the second polynomial times the entire first polynomial. Shift by degree, then add.
🔗 Connection to Binomials
Binomials are polynomials of degree 1. (ax+b)(cx+d) is FOIL. Polynomial multiplication generalizes this: multiply each term in the first by each term in the second, then combine like terms.
(x+2)(x²+3x+1) = x·x² + x·3x + x·1 + 2·x² + 2·3x + 2·1 = x³ + 5x² + 7x + 2
The same distributive principle applies for any number of terms. For three polynomials, multiply the first two, then multiply the result by the third.
📊 Applications: Area and Volume
Area: Rectangle with sides (2x+1) and (3x+2) has area (2x+1)(3x+2) = 6x²+7x+2.
Volume: Box with dimensions (x+1), (x+2), (x+3) has volume (x+1)(x+2)(x+3) = x³+6x²+11x+6.
Polynomial multiplication appears in calculus (product rule), signal processing (convolution), and computer algebra.
⚠️ Disclaimer: This calculator supports polynomials up to degree 3. For higher degrees, the same distributive principle applies. Educational use only.
Polynomial multiplication is the inverse of polynomial division. Use the Multiplying Binomials Calculator for the simpler FOIL case.
Both calculators support the same algebraic patterns from different perspectives.
Expand with multiplication → Factor with division or factoring tools.
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