Polynomial Multiplication Calculator

Multiply polynomials with step-by-step solutions and detailed explanations. Enter coefficients to get instant results with distributive property breakdown.

Calculate Polynomial Multiplication

Multiplication Result

P₁(x) × P₂(x) =

+x^2+3x2

Degree: 2

Step-by-Step Solution

Show detailed steps

Key Concepts

Distributive Property

Each term of the first polynomial multiplies each term of the second polynomial.

Like Terms

Terms with the same power of x are combined by adding their coefficients.

Degree of Result

The degree of the product equals the sum of the degrees of the input polynomials.

Coefficient Format

Enter coefficients as comma-separated values, starting with the constant term.

How to Calculate

Distributive Property Method

P₁(x) × P₂(x) = P₁(x) × (each term of P₂)

Then combine like terms to get the final result

Calculation Steps:

1
Enter polynomial coefficients
Input coefficients separated by commas, starting with constant term
2
Apply distributive property
Multiply each term of first polynomial by each term of second polynomial
3
Combine like terms
Add coefficients of terms with the same power of x
4
Format result
Present the final polynomial in standard form

Important Considerations

⚠️ Input Format

Enter coefficients as comma-separated numbers. Use 0 for missing terms in the middle.

Coefficient Order

Always start with the constant term (degree 0)

• 1,2,3 = 1 + 2x + 3x²
• 0,1,0,2 = x + 2x³

Zero Polynomials

A polynomial with all zero coefficients equals 0

• 0,0,0 = 0
• Any polynomial × 0 = 0

Degree Calculation

Result degree = degree(P₁) + degree(P₂)

• Linear × Linear = Quadratic
• Quadratic × Cubic = Quintic

Large Numbers

Very high degree polynomials may take longer to process

• Consider computational limits
• Break into smaller problems if needed

Example Cases

Case 1: Simple Linear Multiplication

Input: (x + 1) × (x + 2)
Coefficients: 1,1 × 2,1

Result: x² + 3x + 2
Coefficients: 2,3,1

Demonstrates basic distributive property: x·x + x·2 + 1·x + 1·2 = x² + 3x + 2

Case 2: Quadratic × Linear

Input: (x² + 2x + 1) × (x + 3)
Coefficients: 1,2,1 × 3,1

Result: x³ + 5x² + 7x + 3
Coefficients: 3,7,5,1

Shows how each term of the quadratic multiplies each term of the linear polynomial.

Tips

Always double-check your coefficient input format - start with the constant term
Use 0 for missing terms in the middle of a polynomial (e.g., 1,0,3 for 1 + 3x²)
For very long polynomials, consider breaking them into smaller parts
Verify your result by substituting a simple value for x and checking both sides
Remember that the degree of the product is the sum of the input degrees

Frequently Asked Questions

What is polynomial multiplication?

Polynomial multiplication is the process of multiplying two polynomials using the distributive property. Each term of the first polynomial multiplies each term of the second polynomial, and like terms are combined.

How do I enter polynomial coefficients?

Enter coefficients separated by commas, starting with the constant term. For example, "1,2,3" represents 1 + 2x + 3x². The coefficients should be in ascending order of powers.

What is the distributive property?

The distributive property states that a(b + c) = ab + ac. In polynomial multiplication, this means each term of the first polynomial multiplies each term of the second polynomial.

How do I combine like terms?

Like terms have the same power of x. Add their coefficients together. For example, 3x² + 5x² = 8x². This is done after applying the distributive property.

Can I multiply polynomials of any degree?

Yes, this calculator can handle polynomials of any degree. The result will have a degree equal to the sum of the degrees of the input polynomials.