Multiplying Polynomials Calculator
Multiply polynomials using distribution or grid method with detailed solutions
Enter using format: 2x + 3
Enter using format: x - 5
How to Multiply Polynomials
Multiplying polynomials involves using the distributive property to multiply each term of one polynomial by each term of the other. This calculator demonstrates two popular methods: the distributive method and the grid/table method.
The Distributive Method
The distributive property states that a(b + c) = ab + ac. When multiplying polynomials, you distribute each term of the first polynomial across all terms of the second polynomial. This is sometimes called the FOIL method when multiplying two binomials (First, Outer, Inner, Last).
The Grid/Table Method
The grid method provides a visual way to organize polynomial multiplication. Create a table with terms from one polynomial across the top and terms from the other down the side. Fill in each cell with the product of its row and column terms, then add all cells.
Steps to Multiply Polynomials
- Distribute: Multiply each term in the first polynomial by each term in the second
- Apply Exponent Rules: When multiplying terms with the same base, add exponents
- Combine Like Terms: Add coefficients of terms with the same exponent
- Simplify: Write the result in standard form (descending exponents)
Example: Multiplying Polynomials
Let's multiply: (2x + 3)(x - 5)
- First: 2x × x = 2x²
- Outer: 2x × (-5) = -10x
- Inner: 3 × x = 3x
- Last: 3 × (-5) = -15
- Combine: 2x² - 10x + 3x - 15 = 2x² - 7x - 15
Important Rules
- When multiplying variables: x × x = x² (add exponents: x¹ × x¹ = x²)
- When multiplying coefficients: multiply them normally
- Pay attention to signs: negative × negative = positive, negative × positive = negative
- Always combine like terms at the end
Frequently Asked Questions
What is the FOIL method?
FOIL stands for First, Outer, Inner, Last. It's a mnemonic for multiplying two binomials. Multiply the First terms, then the Outer terms, then the Inner terms, and finally the Last terms. Then combine like terms. For example: (a + b)(c + d) = ac + ad + bc + bd.
Does FOIL work for all polynomial multiplication?
FOIL only works for multiplying two binomials (polynomials with exactly two terms each). For polynomials with more terms, you need to use the general distributive method or the grid method, multiplying each term by every other term.
How do I multiply terms with exponents?
When multiplying terms with the same base, add the exponents. For example, x² × x³ = x⁵. Don't forget to also multiply the coefficients: 3x² × 4x³ = 12x⁵. The exponent rule is: xᵃ × xᵇ = x⁽ᵃ⁺ᵇ⁾.
What's the difference between the distributive and grid methods?
Both methods give the same result, but the grid method provides a visual organization that some students find easier. The distributive method is more algebraic and direct. Use whichever method makes more sense to you.
How many terms will the product have?
Before combining like terms, the product will have a number of terms equal to the product of the number of terms in each polynomial. For example, multiplying a trinomial by a binomial gives 3 × 2 = 6 terms initially. After combining like terms, this may reduce.
What about multiplying three or more polynomials?
To multiply three or more polynomials, multiply two at a time. First multiply the first two polynomials, then multiply that result by the third polynomial, and so on. This is the associative property: (A × B) × C = A × (B × C).
How do I avoid mistakes when multiplying polynomials?
Common tips: Write out each step clearly, keep track of negative signs, use the grid method for organization, check that you've multiplied every term by every other term, and always combine like terms at the end. Double-check your arithmetic.
Can I use this for multiplying monomials?
Yes! A monomial is just a polynomial with one term, so all the same rules apply. When multiplying monomials like 3x² and 4x³, multiply the coefficients (3 × 4 = 12) and add the exponents (x² × x³ = x⁵) to get 12x⁵.
Related Calculators
Polynomial Calculator
All polynomial operations in one place
Adding Polynomials Calculator
Add polynomials step-by-step
Subtracting Polynomials Calculator
Subtract polynomials with sign distribution
Polynomial Long Division Calculator
Divide polynomials completely
Synthetic Division Calculator
Quick division for linear divisors
Factor Theorem Calculator
Find polynomial factors