NCERT Class 10 Maths Chapter 2 Polynomials | Full Lecture Notes with Examples, Mnemonics & Worksheets
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NCERT Class 10 Maths – Chapter 2 Polynomials | Full Lecture
NCERT Class 10 Maths – Chapter 2: Polynomials
Full Lecture with Examples, Mnemonics, Diagrams, Worksheets & Animated Slides
Introduction to Polynomials
A polynomial is an algebraic expression consisting of variables (also called indeterminates) and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Example 1: 2x³ + 3x² – 5x + 7 is a polynomial in x.
Example 2: y² – 9y + 20 is a polynomial in y.
Example 3: 5a⁴ – 3a² + a – 6 is a polynomial in a.
Example 4: 3x²y + 2xy² + y² is a polynomial in two variables x and y.
Example 5: –4x³ + 0x² + 7 is still a polynomial (zero coefficients allowed).
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the polynomial.
किसी polynomial का degree उस variable का highest power होता है जो उसमें आता है।
Note: Multiple variables वाले polynomial में degree = सभी variables के powers का sum का सबसे बड़ा मान।
Example 1: Degree of 3x³ + 2x² – 7x + 1 is 3.
Example 2: Degree of y² – 9y + 5 is 2.
Example 3: Degree of 7a⁴ – a³ + 3 is 4.
Example 4: Degree of 6x²y³ – y⁵ is 5.
Example 5: Degree of constant polynomial 9 is 0.
Types of Polynomials
Polynomials can be classified according to the number of terms:
Monomial: One term (e.g., 5x).
Binomial: Two terms (e.g., x² + 5).
Trinomial: Three terms (e.g., x² + 3x + 2).
Quadrinomial: Four terms (rarely used in terminology).
Example 1: 7x is a monomial.
Example 2: 3y² + 2 is a binomial.
Example 3: a² – a + 1 is a trinomial.
Example 4: x³ + 2x² – 3x + 4 is a quadrinomial.
Example 5: –9 is a constant polynomial (monomial).
Zeros of a Polynomial
The zeros of a polynomial are the values of the variable for which the polynomial becomes zero.
वह values of variable जिनसे polynomial का value = 0 हो जाता है।
Example 1: For p(x) = x – 5, zero is x = 5.
Example 2: For p(x) = x² – 9, zeros are x = ±3.
Example 3: For p(y) = y² – 7y + 12, zeros are y = 3, 4.
Example 4: For p(x) = 2x² + 3x, zeros are x = 0, –3/2.
Example 5: For p(a) = a³ – a, zeros are a = 0, 1, –1.
Relation between Zeros and Coefficients
This relation connects the sum and product of zeros with coefficients of the polynomial.
Quadratic Polynomial: ax² + bx + c
Sum of zeros = –b/a, Product of zeros = c/a
यह सबसे important form है।
इसमें हमेशा दो zeros होते हैं (same भी हो सकते हैं या real नहीं भी हो सकते)।
Zeros निकालने के लिए methods:
Factorization
Completing the square
Quadratic formula → x = [–b ± √(b² – 4ac)] / (2a)
Example 1: For x² – 5x + 6, sum = 5, product = 6.
Example 2: For 2x² + 3x – 2, sum = –3/2, product = –1.
Example 3: For x² + 7x + 10, sum = –7, product = 10.
Example 4: For 3x² – 2x – 1, sum = 2/3, product = –1/3.
Example 5: For 4x² + 4x + 1, sum = –1, product = 1/4.
Division Algorithm for Polynomials
For any polynomials p(x) and g(x), we can write: p(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).
Example 1: Divide x² + 3x + 2 by x + 1 → quotient = x + 2, remainder = 0.
Example 2: Divide x³ – 1 by x – 1 → quotient = x² + x + 1, remainder = 0.
Example 3: Divide x³ + 2x² + x + 2 by x + 1 → quotient = x² + x, remainder = 2.
Example 4: Divide x³ – 2x² + 4x – 8 by x – 2 → quotient = x² + 0x + 4, remainder = 0.
Example 5: Divide 2x³ + 3x² + x + 5 by x + 2 → quotient = 2x² – x + 3, remainder = –1.
Applications of Polynomials
Polynomials are used in curve fitting, solving equations, graphing functions, and real-life problem modeling.
Example 1: Profit function of a business can be polynomial.
Example 2: Physics motion equations are polynomials (s = ut + ½at²).
