What Is Multiply Binomials by Polynomials?
To multiply a binomial by a polynomial, multiply each of the binomial's two terms by every term of the polynomial, then add the like terms. A binomial times a trinomial gives six partial products.
Enter a two-term binomial and any polynomial. The calculator shows a table of each partial product, the combined result, its degree and the number of terms. You can also evaluate the product at a value to check your own work.
The degree of the result is the sum of the two degrees. Missing a sign or a term is the most common error, so listing all partial products before combining them is the safest method.
How to Use the Multiply Binomials by Polynomials Calculator
- Enter a two-term binomial.
- Enter the polynomial it multiplies.
- Check the partial products table, then read the combined answer.
Multiply Binomials by Polynomials Formula
Distribution
Degree of the product
Multiply Binomials by Polynomials Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| Binomial | A polynomial with exactly two terms | |
| Polynomial | The expression being multiplied | |
| Partial product | One term times one term, before combining | |
| Degree | Highest power in the result | Sum of the factor degrees |
Multiply Binomials by Polynomials Worked Example
With the example values loaded in the calculator above, the steps are:
- Binomial: 2x + 3. Polynomial: x² − 4x + 5.
- Multiply each term of the binomial by every term of the polynomial (6 partial products, listed in the table).
- Add the partial products and combine like terms: 2x³ − 5x² − 2x + 15.
Result: Product = 2x³ − 5x² − 2x + 15.
Worked example: (2x + 3)(x² − 4x + 5)
Multiply 2x by each term: 2x³ − 8x² + 10x. Multiply 3 by each term: 3x² − 12x + 15.
Add like terms: 2x³ + (−8x² + 3x²) + (10x − 12x) + 15 = 2x³ − 5x² − 2x + 15. At x = 2, the factors are 7 and 1, and the product is 7, which matches 16 − 20 − 4 + 15 = 7.
Checking your answer
Substitute a number into both factors and multiply. The result must match the expanded answer at that same number. It is a quick test for a missed sign or term.
Where Multiply Binomials by Polynomials Is Used
Algebra homework
Check an expansion term by term.
Preparing to factor or solve
Expand products before combining both sides of an equation.
Catching sign errors
Compare the value of the factors with the value of the answer.
Common Multiply Binomials by Polynomials Mistakes to Avoid
- Write all partial products before combining. Combining early is where terms get lost.
- A binomial times a trinomial gives 6 partial products.
Multiply Binomials by Polynomials Terms and Definitions
- Partial product
- The product of one term from each factor, before like terms are combined.
Multiply Binomials by Polynomials FAQs
How do you multiply a binomial by a polynomial?
Distribute each term of the binomial across every term of the polynomial, then add like terms.
How many partial products are there?
Two times the number of terms in the polynomial. A binomial times a trinomial has 6.
What is the degree of the product?
The sum of the degrees of the two factors.
Is this the same as FOIL?
FOIL is the special case of two binomials. Distribution works for any polynomials.
Can I use a different variable?
Yes. Choose it from the variable list.
How do I check my answer?
Substitute any number into both factors and into the result. The values must agree.