Quick Revision Notes: Class 9 Mathematics

Chapter: Polynomials (बहुपद)

### 1. Introduction to Polynomials

• Polynomial: An algebraic expression in which the variables involved have only non-negative integral powers (whole numbers).
  • Example: $2x^3 - 5x^2 + 7x - 3$ is a polynomial in $x$.
  • Non-example: $x + \frac{1}{x}$ (since $\frac{1}{x} = x^{-1}$, and $-1$ is not a whole number).
• Term (पद): The parts of a polynomial separated by '+' or '-' signs.
• Coefficient (गुणांक): The numerical factor of a term.

### 2. Degree of a Polynomial (बहुपद की घात)

• Definition: The highest power of the variable in a polynomial is called its degree.
• Classification based on Degree:
  1. Linear Polynomial (रैखिक बहुपद): Degree = $1$. General form: $ax + b$ (where $a \neq 0$).
  2. Quadratic Polynomial (द्विघात बहुपद): Degree = $2$. General form: $ax^2 + bx + c$ (where $a \neq 0$).
  3. Cubic Polynomial (त्रिघात बहुपद): Degree = $3$. General form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).
  4. Constant Polynomial (अचर बहुपद): Degree = $0$. (e.g., $5, -7$).
  5. Zero Polynomial (शून्य बहुपद): The polynomial $0$. Its degree is not defined.

### 3. Classification based on Number of Terms (पद संख्या के आधार पर)

• Monomial (एकपदी): Polynomial containing $1$ term (e.g., $5x^2$).
• Binomial (द्विपदी): Polynomial containing $2$ terms (e.g., $x + 3$).
• Trinomial (त्रिपदी): Polynomial containing $3$ terms (e.g., $x^2 + 2x + 1$).

### 4. Value of a Polynomial (बहुपद का मान)

• If $p(x)$ is a polynomial in $x$, and $x = a$ is any real number, then the value obtained by putting $x = a$ in $p(x)$ is called the value of $p(x)$ at $x = a$, denoted by $p(a)$.

### 5. Zero of a Polynomial (बहुपद का शून्यक)

• A real number $k$ is said to be a zero of a polynomial $p(x)$ if $p(k) = 0$.
• To find the zero of a linear polynomial $p(x) = ax + b$: $$\text{Set } p(x) = 0 \implies ax + b = 0 \implies x = -\frac{b}{a}$$
  • Note: A non-zero constant polynomial has no zero. Every linear polynomial has one and only one zero.

### 6. Remainder Theorem (शेषफल प्रमेय)

• Let $p(x)$ be any polynomial of degree greater than or equal to $1$ and let $a$ be any real number. If $p(x)$ is divided by the linear polynomial $(x - a)$, then the remainder is $p(a)$.
• For division by $(x + a)$, remainder is $p(-a)$.
• For division by $(ax - b)$, remainder is $p\left(\frac{b}{a}\right)$.

### 7. Factor Theorem (गुणनखंड प्रमेय)

• Let $p(x)$ be a polynomial of degree $n \ge 1$ and $a$ be a real number.
  1. If $p(a) = 0$, then $(x - a)$ is a factor of $p(x)$.
  2. If $(x - a)$ is a factor of $p(x)$, then $p(a) = 0$.

### 8. Algebraic Identities (बीಜगणितीय सर्वसमिकाएँ)

Memorizing these identities is essential for solving factorisation and expansion problems:

1. $(x + y)^2 = x^2 + 2xy + y^2$
2. $(x - y)^2 = x^2 - 2xy + y^2$
3. $x^2 - y^2 = (x + y)(x - y)$
4. $(x + a)(x + b) = x^2 + (a + b)x + ab$
5. $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$
6. $(x + y)^3 = x^3 + y^3 + 3xy(x + y) = x^3 + y^3 + 3x^2y + 3xy^2$
7. $(x - y)^3 = x^3 - y^3 - 3xy(x - y) = x^3 - y^3 - 3x^2y + 3xy^2$
8. $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$
9. $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
10. $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$
    • Special Case: If $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$.