Polynomial Multiplication
Multiply two polynomials — example and distributive law
Polynomial Multiplication Calculator
What is calculated?
This function computes the multiplication of two polynomials using the distributive law. Each term of the first polynomial is multiplied by every term of the second polynomial.
Multiplication info
Distributive law
Multiply each by each:
- Expand the first polynomial
- Each term × every term
- Add exponents: x^a × x^b = x^(a+b)
- Collect like terms
Rule of thumb: When multiplying, exponents add and coefficients multiply.
Quick examples
x × x² = x³
Exponents add
Exponents add
3x × 2x = 6x²
Coefficients multiply
Coefficients multiply
5 × x = 5x
Constant times variable
Constant times variable
(a+b)(c+d) = ac+ad+bc+bd
Distributive
Distributive
FOIL method
First, Outer, Inner, Last - systematic approach for binomial multiplication.
Formulas for polynomial multiplication
Distributive law
\[P(x) \cdot Q(x) = \sum_{i,j} a_i b_j x^{i+j}\]
General multiplication
Binomial multiplication
\[(a+b)(c+d) = ac + ad + bc + bd\]
FOIL method
Power rules
\[x^a \cdot x^b = x^{a+b}\]
Exponents add
Degree of product
\[\deg(P \cdot Q) = \deg(P) + \deg(Q)\]
Degrees add
Step-by-step example
Example: (3x² + 4x + 5) × (2x + 3)
1Write polynomials
\[P(x) = 3x^2 + 4x + 5\]
\[Q(x) = 2x + 3\]
Given polynomials
2Apply distributive law
(3x² + 4x + 5) × (2x + 3)
Multiply each term by each
Multiplication grid
2x
3
Products
3x²
6x³
9x²
→ 6x³ + 9x²
4x
8x²
12x
→ 8x² + 12x
5
10x
15
→ 10x + 15
3Write all terms
\[6x^3 + 9x^2 + 8x^2 + 12x + 10x + 15\]
All individual products
4Collect like terms
x³: 6x³
x²: 9x² + 8x² = 17x²
x¹: 12x + 10x = 22x
x⁰: 15
5Final product
\[P(x) \times Q(x) = 6x^3 + 17x^2 + 22x + 15\]
Final product
FOIL method for binomials
Special case: (a + b)(c + d)
FOIL procedure
- First: Multiply first terms → ac
- Outer: Multiply outer terms → ad
- Inner: Multiply inner terms → bc
- Last: Multiply last terms → bd
Example: (x + 2)(x + 3)
F: x × x = x²
O: x × 3 = 3x
I: 2 × x = 2x
L: 2 × 3 = 6
= x² + 5x + 6
|
|