Class 10 Mathematics | Chapter: Polynomials (बहुपद)

Quick Revision Notes & Formula Sheet

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1. Basic Definitions

• Polynomial (बहुपद): An algebraic expression in the form $p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where the powers of $x$ are non-negative integers.
• Degree (घात): The highest power of the variable $x$ in a polynomial $p(x)$ is called the degree of the polynomial.
• Value of a Polynomial: If $p(x)$ is a polynomial and $k$ is any real number, then the value obtained by replacing $x$ with $k$ is denoted by $p(k)$.

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2. Types of Polynomials (Based on Degree)

1. Linear Polynomial (रैखिक बहुपद): A polynomial of degree 1.
   • General Form: ax + b (where $a \neq 0$)
2. Quadratic Polynomial (द्विघात बहुपद): A polynomial of degree 2.
   • General Form: ax^2 + bx + c (where $a \neq 0$)
3. Cubic Polynomial (त्रिघात बहुपद): A polynomial of degree 3.
   • General Form: ax^3 + bx^2 + cx + d (where $a \neq 0$)

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3. Zeroes of a Polynomial (बहुपद के शून्यक)

• A real number $k$ is said to be a zero of a polynomial $p(x)$ if p(k) = 0.
• Geometrical Meaning: The zeroes of a polynomial $p(x)$ are the x-coordinates of the points where the graph of $y = p(x)$ intersects the X-axis.
  • A Linear polynomial has exactly 1 zero.
  • A Quadratic polynomial can have at most 2 zeroes (Parabola shape).
  • A Cubic polynomial can have at most 3 zeroes.

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4. Relationship between Zeroes and Coefficients

This is the most important part for MP Board Exams.

A. Quadratic Polynomial: ax^2 + bx + c

Let the two zeroes be $\alpha$ (alpha) and $\beta$ (beta).

1. Sum of Zeroes (शून्यकों का योगफल): α + β = -b / a α + β = -(Coefficient of x) / (Coefficient of x^2)

2. Product of Zeroes (शून्यकों का गुणनफल): α * β = c / a α * β = (Constant term) / (Coefficient of x^2)

B. Cubic Polynomial: ax^3 + bx^2 + cx + d

Let the three zeroes be $\alpha$, $\beta$, and $\gamma$ (gamma).

1. Sum of Zeroes: α + β + γ = -b / a
2. Sum of product of zeroes taken two at a time: αβ + βγ + γα = c / a
3. Product of Zeroes: αβγ = -d / a

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5. Formation of a Polynomial

If the zeroes are given, you can form the polynomial using these formulas:

• To form a Quadratic Polynomial: p(x) = x^2 - (Sum of Zeroes)x + (Product of Zeroes) p(x) = x^2 - (α + β)x + (αβ)

• To form a Cubic Polynomial: p(x) = x^3 - (α+β+γ)x^2 + (αβ+βγ+γα)x - (αβγ)

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6. Division Algorithm for Polynomials

If $p(x)$ and $g(x)$ are any two polynomials with $g(x) \neq 0$, then we can find polynomials $q(x)$ and $r(x)$ such that: p(x) = g(x) * q(x) + r(x) Where:

• p(x) = Dividend (भाज्य)
• g(x) = Divisor (भाजक)
• q(x) = Quotient (भागफल)
• r(x) = Remainder (शेषफल)
• Note: Either $r(x) = 0$ or degree of $r(x) <$ degree of $g(x)$.

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7. Important Algebraic Identities (Quick Recap)

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• a^2 - b^2 = (a + b)(a - b)
• x^2 + (a + b)x + ab = (x + a)(x + b)

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Exam Tip: In the MP Board exam, always write the formula before solving the problem to score full step-marking points!