Question

What is polynomial

Answer 1

A polynomial is a mathematical expression that consists of variables (also called indeterminates) and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. In its general form, a polynomial in one variable \( x \) can be expressed as:

\[
P(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_1 x + a_0
\]

where:
- \( P(x) \) is the polynomial,
- \( n \) is a non-negative integer indicating the degree of the polynomial,
- \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants called coefficients,
- \( a_n \neq 0 \) ensures that \( n \) is the highest degree.

Key features of polynomials include:

1. **Degree**: The highest power of the variable in the polynomial. For example, in \( P(x) = 2x^3 + 3x^2 + x + 5 \), the degree is 3.

2. **Coefficients**: The numerical factors in the polynomial. In the example above, the coefficients are 2, 3, 1, and 5.

3. **Terms**: Each part of the polynomial separated by plus or minus signs. The expression \( 2x^3 + 3x^2 + x + 5 \) has four terms.

Polynomials can be classified based on their degree:
- Constant polynomial (degree 0): e.g., \( P(x) = 5 \)
- Linear polynomial (degree 1): e.g., \( P(x) = 2x + 1 \)
- Quadratic polynomial (degree 2): e.g., \( P(x) = x^2 - 3x + 2 \)
- Cubic polynomial (degree 3): e.g., \( P(x) = x^3 + 2x^2 - x + 4 \)
- And so on.

Polynomials can also be in multiple variables, such as \( P(x, y) = 3x^2y + 2xy^2 - y + 1 \).

In many applications across mathematics, science, and engineering, polynomials are used for modeling relationships and solving equations.