Class 9 Mathematic Polynomial All Formula & Concept Notes | Formula Sheet

Understanding polynomials in Class 9 mathematics is foundational for many algebraic concepts you will encounter in higher grades. Here's a detailed overview of the key concepts and formulas related to polynomials:

1. Definition of a Polynomial

• Polynomial: An algebraic expression that consists of variables (also called indeterminates), coefficients, and non-negative integer exponents.
• Standard form: A polynomial in one variable (say $x$) can be expressed as: $$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ where $a_n, a_{n-1}, \ldots, a_1, a_0$ are real numbers and $n$ is a non-negative integer.

2. Types of Polynomials

• Zero Polynomial: $p(x) = 0$
• Constant Polynomial: $p(x) = a_0$ (where $a_0$ is a constant and $n = 0$)
• Linear Polynomial: $p(x) = ax + b$ (where $n = 1$)
• Quadratic Polynomial: $p(x) = ax^2 + bx + c$ (where $n = 2$)
• Cubic Polynomial: $p(x) = ax^3 + bx^2 + cx + d$ (where $n = 3$)

3. Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable in the polynomial.
• Example: For $p(x) = 4x^3 + 2x^2 + x + 7$, the degree is 3.

4. Zeroes of a Polynomial

• The zeroes of a polynomial $p(x)$ are the values of $x$ for which $p(x) = 0$.
• Example: If $p(x) = x^2 - 4$, the zeroes are $x = 2$ and $x = -2$.

5. Addition, Subtraction, and Multiplication of Polynomials

• Addition: Combine like terms (terms with the same powers of the variable).
• Subtraction: Subtract corresponding coefficients of like terms.
• Multiplication: Distribute each term of one polynomial to every term of the other polynomial.

6. Division of Polynomials

• When dividing a polynomial by a non-zero polynomial of a lower degree, the result is expressed as: $$\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}$$
• Example: Divide $2x^3 + 3x^2 + x + 5$ by $x + 1$.

7. Factorization of Polynomials

• Common Factor Method: Taking out the greatest common factor from all terms.
• Factorization by Grouping: Grouping terms with common factors and factoring them.
• Factorization using Identities: Applying algebraic identities to factorize polynomials.

8. Algebraic Identities

• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$
• $a^2 - b^2 = (a + b)(a - b)$
• $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
• $(x + a)(x + b) = x^2 + (a + b)x + ab$

9. Remainder Theorem

• When a polynomial $p(x)$ is divided by $(x - a)$, the remainder is $p(a)$.
• Example: For $p(x) = x^3 - 3x + 2$ and $a = 1$, the remainder is $p(1) = 0$.

10. Factor Theorem

• If $p(a) = 0$, then $(x - a)$ is a factor of $p(x)$.
• Example: For $p(x) = x^3 - 3x + 2$, $x - 1$ is a factor if $p(1) = 0$.

11. Graphical Representation of Polynomials

• The graph of a linear polynomial $ax + b$ is a straight line.
• The graph of a quadratic polynomial $ax^2 + bx + c$ is a parabola.
• The graph of a cubic polynomial $ax^3 + bx^2 + cx + d$ has an "S" shape.

12. Relationship Between Zeroes and Coefficients

• For a quadratic polynomial $ax^2 + bx + c$, if $\alpha$ and $\beta$ are the roots: $$\text{Sum of roots} (\alpha + \beta) = -\frac{b}{a}$$ $$\text{Product of roots} (\alpha \times \beta) = \frac{c}{a}$$

13. Special Cases and Important Notes

• A polynomial of degree $n$ has at most $n$ distinct roots.
• The degree of the zero polynomial is not defined.

These concepts and formulas are essential for understanding polynomials and solving related problems in Class 9 mathematics.