This multiplying polynomials calculator is provided by Hesapstan to multiply two single-variable polynomials and show every partial product before the final result.
What does the multiplying polynomials calculator do?
The multiplying polynomials calculator multiplies every nonzero term in P(x) by every nonzero term in Q(x), then combines like-degree terms into one standard-form polynomial.
It is a general polynomial multiplication tool. It is not limited to binomial × binomial problems.
Multiply all term pairs first, then collect products with the same power of x.
How polynomial multiplication works
Polynomial multiplication is the distributive property applied repeatedly. Each term of the first polynomial must be distributed across every term of the second polynomial.
- Take one term from P(x).
- Multiply it by every term of Q(x).
- Repeat for all terms of P(x).
- Group products that have the same degree.
- Add their coefficients to get the standard-form result.
The calculator displays the partial products so you can see what was multiplied before the final combination step.
This is broader than FOIL
FOIL is a shortcut for multiplying two binomials only. A general polynomial may have more than two terms, so FOIL is not enough.
- FOIL is useful for (ax+b)(cx+d).
- Box or area methods are visual tools for selected forms.
- This calculator handles the general n-term polynomial multiplication pattern.
Use this calculator when the product is not just a binomial × binomial exercise.
Supported input format
Enter P(x) and Q(x) as single-variable polynomials such as 3x^2 - 2x + 1.
- The variable is x.
- The maximum supported degree is 8.
- Multivariable terms are not supported.
- Named functions, parentheses as general expressions, and fractional exponents are rejected.
- A zero polynomial is allowed and returns 0.
The multiplication algorithm is mechanical, but a high-degree product creates a large partial-products list. The cap keeps the result readable on mobile and desktop.
Worked example
For P(x)=2x^2−x+3 and Q(x)=x−4, each term in P(x) is multiplied by x and by −4.
- 2x^2 × x = 2x^3
- 2x^2 × −4 = −8x^2
- −x × x = −x^2
- −x × −4 = 4x
- 3 × x = 3x
- 3 × −4 = −12
Combining like-degree terms gives 2x^3−9x^2+7x−12.
Why like terms must be combined
Like terms have the same power of x. Products such as −8x^2 and −x^2 both belong to degree 2, so their coefficients are added.
Without this step, the expanded expression may list all products correctly but is not yet in standard form.
A standard polynomial is usually written from highest degree down to constant term.
Zero and missing terms
Missing degrees behave as zero-coefficient terms. You do not need to type them explicitly unless it helps you read the expression.
If one input is the zero polynomial, the product is 0. The calculator treats this as a valid result, not as an error.
Common mistakes and limitations
- Multiplying only the first and last terms and missing inner term pairs.
- Losing negative signs during term multiplication.
- Forgetting to combine like-degree terms.
- Trying to apply FOIL to polynomials with more than two terms.
- Expecting factorization; multiplication expands, it does not factor.
This calculator is for single-variable polynomials. It does not parse general symbolic algebra, multivariable products, or named functions.
Frequently Asked Questions
Is multiplying polynomials the same as FOIL?
No. FOIL is only a binomial shortcut. This calculator handles general single-variable polynomial products.
What is a partial product?
A partial product is the result of multiplying one term from the first polynomial by one term from the second polynomial.
Why combine like terms?
Terms with the same power of x represent the same degree, so their coefficients must be added in the final standard form.
Does the calculator factor polynomials?
No. It expands a product. Factoring is the reverse task and belongs to other calculators.
Why is there a degree limit?
The degree limit keeps the partial-products display readable and prevents an oversized result list.