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All important formula regarding this chapter for iit Madras qualifier (BS in data science and application)

Based on the contents of the provided outline, here are some key formulas and concepts relevant to polynomial functions for the IIT Madras qualifier in BS in Data Science and Applications:

### 4.1 Definition of Polynomial Function
A polynomial function is a mathematical expression consisting of variables (x) and coefficients (a), combined using addition, subtraction, and multiplication. It has the general form:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]
where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(n\) is a non-negative integer.

### 4.2 Classification of Polynomials
#### 4.2.1 Based on the Number of Variables
- **Univariate Polynomial:** Involves one variable, e.g., \(P(x) = x^2 + 3x + 2\).
- **Multivariate Polynomial:** Involves more than one variable, e.g., \(P(x, y) = x^2 + xy + y^2\).

#### 4.2.2 Based on the Degree of Polynomial
- **Degree:** The highest power of the variable in the polynomial.
  - Example: For \(P(x) = 4x^3 + 3x^2 + x + 1\), the degree is 3.
- **Linear Polynomial:** Degree 1, e.g., \(P(x) = ax + b\).
- **Quadratic Polynomial:** Degree 2, e.g., \(P(x) = ax^2 + bx + c\).
- **Cubic Polynomial:** Degree 3, e.g., \(P(x) = ax^3 + bx^2 + cx + d\).

#### 4.2.3 Based on the Number of Terms
- **Monomial:** One term, e.g., \(P(x) = ax^n\).
- **Binomial:** Two terms, e.g., \(P(x) = ax^n + bx^m\).
- **Trinomial:** Three terms, e.g., \(P(x) = ax^n + bx^m + cx^k\).

### 4.3 Operations on Polynomial Functions
#### 4.3.1 Addition of Polynomials
\[ (a_n x^n + \cdots + a_1 x + a_0) + (b_n x^n + \cdots + b_1 x + b_0) \]
Combine like terms.

#### 4.3.2 Subtraction of Polynomials
\[ (a_n x^n + \cdots + a_1 x + a_0) - (b_n x^n + \cdots + b_1 x + b_0) \]
Combine like terms.

#### 4.3.3 Multiplication of Polynomials
\[ (a_m x^m + \cdots + a_0) \cdot (b_n x^n + \cdots + b_0) \]
Use distributive property to multiply each term in the first polynomial by each term in the second polynomial.

#### 4.3.4 Division of Polynomials
- **Long Division:** Similar to arithmetic long division.
- **Synthetic Division:** A simplified method used when the divisor is a linear polynomial.

### 4.4 Characteristics of Polynomial Functions
- **Roots/Zeros:** Solutions of \(P(x) = 0\).
- **End Behavior:** Determined by the leading term \(a_n x^n\).
  - If \(a_n > 0\) and \(n\) is even, \(P(x) \to +\infty\) as \(x \to \pm\infty\).
  - If \(a_n > 0\) and \(n\) is odd, \(P(x) \to +\infty\) as \(x \to +\infty\) and \(P(x) \to -\infty\) as \(x \to -\infty\).

### 4.5 Zeroes of Polynomial Functions
- **Factor Theorem:** \(x - c\) is a factor of \(P(x)\) if and only if \(P(c) = 0\).
- **Rational Root Theorem:** Provides a possible list of rational roots of \(P(x) = a_n x^n + \cdots + a_1 x + a_0\).

Would you like further details or explanations on any of these topics? Here are some additional questions you might consider asking:

1. How do you perform polynomial long division?
2. Can you explain synthetic division with an example?
3. What are the specific end behaviors for polynomials of various degrees and leading coefficients?
4. How do you determine the number and multiplicity of roots for a given polynomial?
5. Can you explain the process of finding the zeroes of a polynomial function using the Rational Root Theorem?

**Tip:** When solving polynomial equations, always check for possible factoring first, as it can simplify the solution process significantly.

Explore essential formulas and concepts for polynomial functions required for the IIT Madras qualifier in BS Data Science and Applications. Topics include classification, operations, characteristics, and finding zeroes of polynomial functions.

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