Multiplying Polynomials Calculator
Multiply two polynomials with step-by-step solutions and detailed explanations
Polynomial Multiplication Calculator
Formula
First Polynomial P(x)
Highest power of x in the polynomial
Coefficients (from highest to lowest degree):
P(x) =
Second Polynomial Q(x)
Highest power of x in the polynomial
Coefficients (from highest to lowest degree):
Q(x) =
Result
Result Coefficients
Step-by-Step Solution
Step 1: Original Problem
Step 2: Apply Distributive Property
Step 3: Individual Term Multiplications
Step 4: Combine Like Terms
Step 5: Final Answer
Example Problems
Example 1: Multiplying Binomials
Problem: Multiply (x + 2)(x + 3)
Solution:
- x × x = x²
- x × 3 = 3x
- 2 × x = 2x
- 2 × 3 = 6
Combine like terms: x² + 3x + 2x + 6 = x² + 5x + 6
Example 2: Polynomial × Binomial
Problem: Multiply (x² - 2x + 1)(x + 4)
Solution:
Multiply each term in the first polynomial by each term in the second:
- x² × x = x³
- x² × 4 = 4x²
- (-2x) × x = -2x²
- (-2x) × 4 = -8x
- 1 × x = x
- 1 × 4 = 4
Combine like terms: x³ + 4x² - 2x² - 8x + x + 4 = x³ + 2x² - 7x + 4
Example 3: Two Trinomials
Problem: Multiply (x² + x + 1)(x² - x + 1)
Solution:
This is a special case that results in:
Result: x⁴ - x² + x² + 1 = x⁴ + 1
Note: Many middle terms cancel out in this symmetric case
Multiplication Rules
Distributive Property
Each term in the first polynomial multiplies each term in the second polynomial
Combine Powers
When multiplying terms with the same variable, add the exponents: x^a × x^b = x^(a+b)
Combine Like Terms
After multiplication, add coefficients of terms with the same power
Degree Rule
The degree of the product equals the sum of the degrees of the factors
Key Concepts
Monomial
A single term (e.g., 3x², -5, 2xy)
Binomial
Two terms (e.g., x + 2, 3x² - 5x)
Trinomial
Three terms (e.g., x² + 3x + 2)
Polynomial
Expression with multiple terms involving variables with non-negative integer exponents
Degree
Highest power of the variable in the polynomial
Quick Tips
Use the distributive property: multiply each term by every other term
Add exponents when multiplying variables: x² × x³ = x⁵
Multiply coefficients normally: 3x × 4x = 12x²
Always combine like terms in the final answer
Write answers in descending order of powers
Understanding Polynomial Multiplication
What are Polynomials?
A polynomial is an algebraic expression consisting of variables and coefficients, involving only non-negative integer powers of variables. Examples include 3x² + 2x - 5 and x⁴ - x² + 1.
The Distributive Property
When multiplying polynomials, we apply the distributive property: each term in the first polynomial must be multiplied by each term in the second polynomial.
(a + b)(c + d) = ac + ad + bc + bd
Step-by-Step Process
Follow these systematic steps to multiply any two polynomials:
Apply Distributive Property
Multiply each term in first polynomial by each term in second
Multiply Terms
Multiply coefficients and add exponents for like variables
Combine Like Terms
Add coefficients of terms with same variable powers
Arrange in Standard Form
Write result in descending order of powers
Special Cases
Monomial × Polynomial
Multiply the monomial by each term in the polynomial
3x(x² + 2x - 1) = 3x³ + 6x² - 3x
Binomial × Binomial
Use FOIL method or distributive property
(x+a)(x+b) = x² + (a+b)x + ab
General Case
Apply distributive property systematically
Result degree = sum of input degrees