Key Concept

Basic Fundamentals
  • The degree of polynomial : the highest power of variable in any equation.
  • Linear polynomial : if degree of polynomial  =1  , for ex. 4x + 2 is a polynomial equation in the variable x of degree 1
  • Quadratic polynomial : if degree of polynomial =2  , for ex. 2y2 – 3y + 4 is a polynomial in the variable y of degree 2
  • Cubic polynomial : if degree of polynomial =3  , for ex. 2y3 – 5y + 6 is a polynomial in the variable y of degree 3
  • Zero of a polynomial :k is said to be zero of a polynomial p(x) if  p(k) = 0 , where k is a real number.
  • Relation between the zeros  of polynomials and coefficients of a quadratic  polynomial:
    • For any polynomial P(x) = ax2 + bx + c if the zeroes of the quadratic polynomial are α, and β then,
      • (α + β) = -b/a
      • αβ =c/a
    • For any polynomial P(x) = ax3 +bx2 + cx + d if the zeroes of the quadratic polynomial are α, β and  γ  then,
      • (α + β+ γ ) = -b/a
      • αβ+βγ+γα =  c/a
      • αβγ  =-d/a
  • If α, β are roots of a quadratic polynomial p(x), then p(x) = x2 – (α + β) x + αβ
    • ⇒ p(x) = x2 – (sum of roots) x + product of roots
  • If α, β and γ are zeroes of a cubic polynomial p(x),Then, p(x) =x3 – (α + β + γ) x2 + (αβ + βγ + αγ) x – (αβγ)
                  ⇒ p(x) = x3 – (sum of zeroes) x2 + (sum of product of zeroes / roots taken two at a time)x – (product of zeroes)

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