Multiplying Polynomials Calculator (expand & simplify the product)

Understanding the multiplying polynomials calculator — concept and principle

This calculator multiplies two polynomials in x and simplifies the product. It tidies both inputs, multiplies every term by every term, combines like terms and reports the degree, the number of terms and the sum of the coefficients as a quick check. Use it for algebra homework and to check expansions.

How it is calculated

Every term of the first polynomial times every term of the second: c·xⁱ × d·xʲ = cd·x^(i+j)

Combine like terms and write the result from the highest power down.

• Degree of the product = sum of the two degrees.
• Check: the sum of the product's coefficients equals the product of the two coefficient sums (the value at x = 1).

The principle

Multiplying polynomials is the distributive property applied term by term: each term of one polynomial multiplies the whole of the other. Coefficients multiply and exponents add because xⁱ·xʲ = x^(i+j). For two binomials this is the familiar FOIL pattern; for longer polynomials the box method organises the same products in a grid. Putting x = 1 turns every polynomial into the sum of its coefficients, giving a quick check.

Worked example

Default inputs: (2x² + 3x − 1)(x − 4).

1. 2x² × (x − 4) = 2x³ − 8x².
2. 3x × (x − 4) = 3x² − 12x.
3. −1 × (x − 4) = −x + 4.
4. Combine: 2x³ + (−8 + 3)x² + (−12 − 1)x + 4 = 2x³ − 5x² − 13x + 4.
5. Degree 3, 4 terms; sum of coefficients = 2 − 5 − 13 + 4 = −12, and (2 + 3 − 1)(1 − 4) = 4 × (−3) = −12.

How to read the result

The leading term (2x³) is the product of the two leading terms, and the constant (4) is the product of the two constants, which are quick spot checks. The degree rises by adding degrees: a quadratic times a linear polynomial is a cubic.

Common mistakes

• Multiplying only some pairs of terms.
• Multiplying exponents instead of adding them.
• Losing a negative sign, especially with −1 × −4.
• Combining terms with different powers.

Frequently asked questions

How do you multiply polynomials?

Multiply each term of the first by each term of the second, then combine like terms.

What is the degree of a product of polynomials?

The sum of the degrees of the factors.

How can I check the product quickly?

Substitute x = 1 (or another value) into the factors and the product and compare.

Sources: OpenStax, Elementary Algebra 2e (2020), §6.3 Multiply Polynomials