Multiplying Polynomials Calculator

Understanding Polynomial Multiplication

Multiplying polynomials is one of the core foundational skills in algebra. A polynomial is an algebraic expression composed of variables, coefficients, and non-negative integer exponents linked by addition, subtraction, and multiplication. When multiplying two or more polynomials, the fundamental rule is that every term of the first polynomial must be multiplied by every term of the second polynomial, after which all resulting like terms are combined.

The product of two polynomials $P(x)$ and $Q(x)$ is denoted as:

$$R(x) = P(x) \cdot Q(x)$$

If the degree of $P(x)$ is $m$ and the degree of $Q(x)$ is $n$, then the degree of their product $R(x)$ is always the sum of their individual degrees:

$$\deg(P \cdot Q) = \deg(P) + \deg(Q) = m + n$$

Three Methods for Multiplying Polynomials

Depending on the number of terms and your personal preference, there are three primary techniques for computing polynomial products:

1. The Distributive Property and FOIL Method

The distributive property states that $a(b + c) = ab + ac$. When multiplying larger polynomials, this principle extends across all terms. For two binomials $(a + b)(c + d)$, this is commonly remembered using the FOIL acronym:

• F (First): Multiply the first terms in each set of parentheses: $a \cdot c$.
• O (Outer): Multiply the outermost terms: $a \cdot d$.
• I (Inner): Multiply the innermost terms: $b \cdot c$.
• L (Last): Multiply the last terms in each set of parentheses: $b \cdot d$.

For binomials, you can also explore our dedicated FOIL Method Calculator.

2. The Box / Grid Method (Area Model)

The Box Method (also called the geometric area model) organizes polynomial multiplication into a visual grid where the terms of the first polynomial form the rows and the terms of the second polynomial form the columns. Each grid cell contains the product of its corresponding row and column:

$$\text{Cell}(i, j) = (a_i x^i) \cdot (b_j x^j) = (a_i \cdot b_j) x^{i+j}$$

After populating the table, terms sharing identical powers of $x$ (often situated along diagonals) are summed together. This approach virtually eliminates missing term errors and sign mistakes when working with trinomials and higher-order polynomials. You can also explore our Generic Rectangle Calculator for additional geometric factoring models.

3. The Column / Vertical Method

Similar to multi-digit arithmetic multiplication, the vertical method places one polynomial above the other. Each term of the bottom polynomial is multiplied by the entire top polynomial, and partial products are aligned in columns according to their variable powers for straightforward vertical addition.

Step-by-Step Multiplication Example

Consider multiplying the binomial $P(x) = 2x + 3$ by the trinomial $Q(x) = x^2 - 4x + 5$:

1. Distribute $2x$: $$2x \cdot (x^2 - 4x + 5) = 2x^3 - 8x^2 + 10x$$
2. Distribute $+3$: $$3 \cdot (x^2 - 4x + 5) = 3x^2 - 12x + 15$$
3. Combine all terms: $$(2x^3 - 8x^2 + 10x) + (3x^2 - 12x + 15)$$
4. Group like terms: $$2x^3 + (-8x^2 + 3x^2) + (10x - 12x) + 15$$
5. Write in standard form: $$2x^3 - 5x^2 - 2x + 15$$

Special Polynomial Multiplication Formulas

Certain binomial products appear frequently and follow memorable algebraic patterns:

• Difference of Squares: $(a + b)(a - b) = a^2 - b^2$
• Square of a Sum: $(a + b)^2 = a^2 + 2ab + b^2$
• Square of a Difference: $(a - b)^2 = a^2 - 2ab + b^2$
• Cube of a Binomial: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

For addition and subtraction of polynomials, check out our Add and Subtract Polynomials Calculator, or use our Expand Polynomials Calculator to expand higher algebraic powers.