Multiplying polynomials is the distributive property applied many times: every term of the first polynomial multiplies every term of the second, and like terms are combined. The degree of the product is the sum of the degrees.
How to use the Multiplying Polynomials Calculator
- Choose the degree of each polynomial – the matching coefficient fields appear.
- Enter the coefficients from the highest power down; leave a field at 0 for a missing power.
- Read the result, written with powers of x in descending order.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator convolves the coefficient lists – the coefficient of xⁿ in the product is the sum of aᵢbⱼ over all i + j = n.
Tips
- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.
Common mistakes to avoid
- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.
Formula
(Σ aᵢxⁱ)(Σ bⱼxʲ) = Σ (aᵢbⱼ) x^(i+j)Worked example
Inputs — Degree of P(x): 3; a₃ (x³): 1; a₂ (x²): 2; a₁ (x): 0; a₀: -1; Degree of Q(x): 2; b₂ (x²): 1; b₁ (x): -3; b₀: 2.
Results — P(x) × Q(x) = x⁵ − x⁴ − 4x³ + 3x² + 3x − 2; P(x) = x³ + 2x² − 1; Q(x) = x² − 3x + 2; Degree of the product = 5.
Key terms
- Distributive property
- a(b + c) = ab + ac.
Frequently asked questions
What is the degree of a product?
The sum of the degrees: a cubic times a quadratic is degree 5.
How many terms before simplifying?
The number of terms of P times the number of terms of Q.
Is polynomial multiplication commutative?
Yes, P·Q = Q·P.
What is the fastest method by hand?
Use a grid (table) or vertical multiplication like long multiplication.