CH02 : Polynomials CBSE Class 10th

Polynomials - Class 10 Maths | Pratap Sanjay Sir | CBSE | NCERT

Polynomial

A polynomial in which the power of the variable is a whole number.

Examples: π‘₯+2π‘₯, π‘₯3+π‘₯2+3π‘₯+4, π‘₯3+3,6 etc.

Types of Polynomials

1. On the basis of power:

  • Linear Polynomial (eg- 2x+y+4)
  • Quadratic Polynomial (eg- 3x2-8x+9)
  • Cubic Polynomial (eg- 4x3-x2+6x+3)

2. On the basis of terms:

  • Monomial
  • Binomial
  • Trinomial

Fundamentals

An algebraic equation of the form:

p(x) = a₀xⁿ + a₁xⁿ⁻¹ + ... + an-1x + an

is called a polynomial, provided it has no negative exponent for any variable.

Where, a₀, a₁, ..., an are constants (real numbers); a₀ ≠ 0.

Degree of Polynomial

n is called the degree (highest power of variable x):

  • Linear polynomial: ax + b (where, a ≠ 0)
  • Quadratic polynomial: ax² + bx + c (where, a ≠ 0)
  • Cubic polynomial: ax³ + bx² + cx + d (where, a ≠ 0)

Zeroes of Polynomial

For polynomial p(x), the value of x for which p(x) = 0, is called zero(es) of the polynomial.

  • Linear polynomial can have at most 1 root (zero).
  • Quadratic polynomial can have at most 2 roots (zeroes).
  • Cubic polynomial can have at most 3 roots.

Relationship between Zeroes and Coefficients

  • Zero of linear polynomial ax + b is given by x = -𝑏 / π‘Ž
  • If Ξ± and Ξ² are zeroes of the quadratic polynomial ax² + bx + c:
    • Sum of zeroes, Ξ± + Ξ² = -𝑏 / π‘Ž
    • Product of zeroes, Ξ±Ξ² = 𝑐 / π‘Ž
  • If Ξ±, Ξ², and Ξ³ are zeroes of the cubic polynomial ax³ + bx² + cx + d:
    • Sum of zeroes, Ξ± + Ξ² + Ξ³ = -𝑏 / π‘Ž
    • Sum of product of pairs, Ξ±Ξ² + Ξ²Ξ³ + Ξ³Ξ± = 𝑐 / π‘Ž
    • Product of zeroes, 𝛼𝛽𝛾 = -𝑑 / π‘Ž

Division Algorithm for Polynomials

If p(x) and g(x) are two polynomials with g(x) ≠ 0, then:

p(x) = g(x) x q(x) + r(x)

where, Dividend = Divisor x Quotient + Remainder.

Tips

  • Graph of linear equation is a straight line, while graph of quadratic equation is a parabola.
  • Degree of polynomial = maximum number of zeroes of polynomial.
  • If remainder r(x) = 0, then g(x) is a factor of p(x).
  • To form a quadratic polynomial, if sum and product of zeroes are given: P(x) = x² - (sum of zeroes)x + (product of zeroes)

Graphs

Graph of Linear Polynomial

Example: Y = x + 2

Graph of Linear Polynomial

Graph of a linear polynomial is a straight line which intersects the x-axis at one point, indicating a degree of 1.

Graph of Quadratic Polynomial

Examples:

  • Case 1: Graph cuts the x-axis at 2 points, indicating 2 zeroes.
  • Case 2: Graph touches the x-axis at 1 point, indicating a perfect square.
  • Case 3: Graph does not intersect the x-axis, indicating no real zeroes.
Graph of Quadratic Polynomial

NCERT Exercises

Exercise 2.1

Question 1: Find the zeroes of the quadratic polynomial ( x^2 + 7x + 10 ), and verify the relationship between the zeroes and the coefficients.

To find the zeroes of ( x^2 + 7x + 10 ):

  • Factorize the polynomial: ( x^2 + 7x + 10 = (x+5)(x+2) ).
  • Set each factor to zero: ( x+5=0 ) or ( x+2=0 ).
  • Solve for ( x ): ( x=-5 ) or ( x=-2 ).

The zeroes of ( x^2 + 7x + 10 ) are -5 and -2.

Verification:

  • Sum of zeroes: ( -5 + (-2) = -7 ), which is equal to ( -b/a ) (coefficient of ( x ) divided by coefficient of ( x^2 )).
  • Product of zeroes: ( -5 * (-2) = 10 ), which is equal to ( c/a ) (constant term divided by coefficient of ( x^2 )).

Previous Year's Question

1. If the sum of zeroes of the quadratic polynomial 3x² - Kx + 6 is 3, then find the value of K.