Welcome to the **Multiplying Polynomials Calculator**. This tool simplifies the process of multiplying any two polynomials, converting them into standard form and providing a step-by-step solution. Enter your expressions below and get the expanded product instantly.
Multiplying Polynomials Calculator
Product Polynomial
Detailed Calculation Steps
Multiplying Polynomials Formula
Polynomial multiplication is defined by the distributive property. If we have two polynomials $P(x)$ and $Q(x)$, their product $R(x)$ is calculated as:
If $P(x) = \sum_{i=0}^{n} a_i x^i$ and $Q(x) = \sum_{j=0}^{m} b_j x^j$, then the product $R(x) = P(x) \cdot Q(x)$ is:
$$R(x) = \sum_{k=0}^{n+m} c_k x^k$$
where the coefficient $c_k$ is the convolution of $a_i$ and $b_j$:
$$c_k = \sum_{i=0}^{k} a_i b_{k-i}$$
This is essentially applying the distributive property (FOIL method for binomials) across all terms and summing terms with the same exponent.
Variables Explained
The calculator uses two primary string variables for input:
- Polynomial 1 ($P(x)$): The first polynomial expression to be multiplied. Accepts standard notation (e.g., coefficients, $x$, and exponents using $^$).
- Polynomial 2 ($Q(x)$): The second polynomial expression.
- Product Polynomial ($R(x)$): The resulting polynomial after multiplying $P(x)$ and $Q(x)$.
What is Multiplying Polynomials?
Polynomial multiplication is a fundamental algebraic operation. It involves taking two polynomial expressions and multiplying every term in the first polynomial by every term in the second polynomial. The process concludes by combining all resulting terms that have the same variable exponent (like terms).
For simple binomials, this is commonly known as the **FOIL method** (First, Outer, Inner, Last). For polynomials with more terms, the systematic application of the distributive property ensures that all possible pairings are multiplied, leading to the final product in its simplest, standard form.
The degree of the resulting product polynomial is always the sum of the degrees of the original two polynomials.
How to Calculate Multiplying Polynomials (Example)
Let’s find the product of $P(x) = (3x – 1)$ and $Q(x) = (x^2 + 2x + 4)$.
- Apply the Distributive Property: Multiply each term in $P(x)$ by every term in $Q(x)$. $$ (3x – 1)(x^2 + 2x + 4) = 3x(x^2 + 2x + 4) – 1(x^2 + 2x + 4) $$
- Expand the Products: $$ 3x(x^2) + 3x(2x) + 3x(4) – 1(x^2) – 1(2x) – 1(4) $$ $$ 3x^3 + 6x^2 + 12x – x^2 – 2x – 4 $$
- Combine Like Terms: Group terms by their exponent.
- $x^3$ terms: $3x^3$
- $x^2$ terms: $6x^2 – x^2 = 5x^2$
- $x$ terms: $12x – 2x = 10x$
- Constant terms: $-4$
- Final Result: Write the simplified polynomial in standard form. $$ R(x) = 3x^3 + 5x^2 + 10x – 4 $$
Frequently Asked Questions (FAQ)
What is the standard form of a polynomial?
Standard form means the terms are written in descending order of the exponents, from the highest degree term down to the constant term.
What happens if a polynomial has only one term?
A polynomial with only one term is called a monomial. The multiplication rules still apply: multiply the coefficients, and add the exponents of the variables.
Does the order of multiplication matter?
No, polynomial multiplication is commutative, meaning $P(x) \cdot Q(x)$ is equal to $Q(x) \cdot P(x)$. The order of input does not change the final product.
What are the requirements for polynomial input?
The calculator requires standard notation: integer or decimal coefficients, $x$ as the variable, and $^$ for exponents (e.g., $5x^3 – 2.5x + 1$).