Multiplying Polynomials Calculator - FOIL and Convolution

A multiplying polynomials calculator is a single tool that takes two single-variable polynomials, P(x) and Q(x), and returns their expanded product (P * Q)(x) in standard form. Type the coefficients of P(x) and Q(x) from the highest power of x down to the constant term, and the result panel shows the standard-form product, its highest degree, the count of non-zero terms, and the list of partial products that combine to give each result term.

• • Homework and textbook check: Verify the product of two polynomials without redoing the FOIL step or the general convolution by hand.
• • Expand a binomial times a trinomial: Compute the product (ax + b)(cx^2 + dx + e) where FOIL no longer applies.
• • Square a polynomial: Use the same calculator to compute P(x)^2, which is P(x) multiplied by itself.
• • Prepare inputs for follow-up tools: Generate the standard-form product polynomial that will be the dividend for the related polynomial tools.

Multiplying polynomials is the algebra operation that turns a factored expression into an expanded one. The product inherits the same variable and the highest degree becomes the sum of the two inputs' highest degrees.

When the same two polynomials need to be added or subtracted instead, our Add Subtract Polynomials Calculator pairs the coefficients and writes the result in standard form.

The calculator builds the convolution of the two coefficient sequences, then writes the surviving terms in standard form. Every term in P(x) is multiplied by every term in Q(x), and the exponents of x are added.

• P(x): The first polynomial, with coefficients a3, a2, a1, a0 for x^3 down to the constant term.
• Q(x): The second polynomial, with coefficients b3, b2, b1, b0 for the same powers of x.
• a_i, b_j: The coefficient of x^i in P(x) and the coefficient of x^j in Q(x) respectively. Zero when the polynomial has no term of that power.
• c_k: The coefficient of x^k in the product, equal to the sum of a_i * b_j over all i + j = k. This is the discrete convolution of the two coefficient sequences at offset k.

The convolution runs over every power of x from 0 up to 6 and collects all a_i * b_j partial products whose exponents add to that power. Zero partial products are dropped before display, so the partial products list is shorter than 16 whenever a coefficient is zero.

According to OpenStax, multiplying polynomials is built up from monomial times monomial, to binomial times binomial with FOIL, to the general case where every term of one factor multiplies every term of the other before like terms combine.

If both inputs are binomials, our Multiplying Binomials Calculator shows the four FOIL partial products and the final quadratic on a more compact panel.

Four small ideas make the calculator's work predictable and explain why the result can have more or fewer terms than the number of partial products.

These four ideas are the entire rule set. The rest of the calculator is just enumerating the partial products, summing the coefficients that share a power of x, and presenting the surviving terms in the conventional order.

Once the product is in standard form, the dividend for our Polynomial Division Calculator is the result polynomial.

Type the two polynomials as coefficients from the highest power down to the constant term, and read the product polynomial, highest degree, non-zero term count, and partial products list on the right.

1. 1 Enter the coefficients of P(x): Type x^3, x^2, x, and the constant term in the four P(x) fields. Use 0 for any missing term.
2. 2 Enter the coefficients of Q(x): Type x^3, x^2, x, and the constant term in the four Q(x) fields.
3. 3 Read the product polynomial: Look at the highlighted Product Polynomial row for the standard-form product.
4. 4 Check the highest degree and term count: Use the Result Highest Degree and Non-Zero Terms rows to confirm the structure.
5. 5 Inspect the partial products: Use the Partial Products row to see every a_i * b_j * x^(i+j) term, so you can verify the convolution.
6. 6 Adjust and re-read: Change any coefficient and the product, highest degree, term count, and partial products list update together.

Example: a student is asked to multiply (2x + 1)(x^2 - 3x + 4). They type 2 in P(x) x, 1 in P(x) constant, 1 in Q(x) x^2, -3 in Q(x) x, and 4 in Q(x) constant. They read 2x^3 - 5x^2 + 5x + 4 in the Product Polynomial row, with a Highest Degree of 3 and 4 non-zero terms. The Partial Products row shows 2x^3 - 6x^2 + 8x + x^2 - 3x + 4, which is the list of a_i * b_j * x^(i+j) terms that combine into the standard-form result.

Once the product polynomial is in standard form, our Polynomial Graphing Calculator plots it on the same axes as P(x) and Q(x).

The bookkeeping is the kind of arithmetic that breaks in long homework sets or multi-step modeling problems, especially when the inputs have more than two terms.

• • One pass for all partial products: Type the four coefficients of P(x) and the four of Q(x) once, and the calculator enumerates every a_i * b_j * x^(i+j) term at the same time.
• • Standard form without sign mistakes: The result string is built from the highest non-zero power down to the constant term with a plus or minus between every term, and zero terms are dropped automatically.
• • Highest degree and term count visible together: The result panel reports the highest degree and the non-zero term count next to the polynomial string.
• • Works with decimals and negatives: Coefficient inputs such as 0.5, -1.25, and -3 are kept as exact values, then formatted with trailing zeros trimmed.
• • FOIL and convolution in one place: The same calculator reduces to FOIL when both inputs are binomials.

The biggest practical benefit is that the partial products list is rendered explicitly. A student can show the work by reading the partial products row, while a tutor can scan the convolution to see which a_i * b_j terms combined to give each result coefficient.

If the product polynomial ends up as a quadratic with three non-zero terms, our Factoring Trinomials Calculator factors the trinomial so the product can be written back in factored form.

The convolution is fixed, but the value of the coefficients and which coefficients are zero change the result.

• • The calculator accepts two polynomials in one variable, x. Polynomials in two or more variables such as x^2 + xy + y^2 are not supported.
• • The input panel handles polynomials up to degree 3 in each of P(x) and Q(x), so the product is at most degree 6. Higher-degree inputs have to be split.

For most algebra work the limitation is a non-issue, because the common textbook case is a single-variable polynomial of degree 2 or 3. When the result is a quadratic, the factoring trinomials calculator picks up the next step.

According to Wolfram MathWorld, the product of two polynomials in one indeterminate is itself a polynomial whose coefficient of x^k equals the convolution of the two coefficient sequences at offset k.

According to Paul's Online Math Notes, multiplying two polynomials uses the distributive property so that every term in the first polynomial is multiplied by every term in the second, then like terms are collected.

When the result of the multiplication is a degree 2 polynomial with three non-zero terms, our Quadratic Formula Calculator solves the equation form of that quadratic and returns its real roots.