|Class 10 Maths Polynomials||Relationship Zeroes & Coefficients of Polynomia|
Relationship between Zeroes and Coefficients of a Polynomial
We know that zero of a linear polynomial ax + b is –b/a
Let’s find the relationship between zeroes and coefficients of a quadratic polynomial.
For a quadratic polynomial p(x) = ax2 + bx + c, if the roots are α & β, then α + β = -b/a & α * β = c/a
For Proof, refer ExamFear video lessons on this chapter.
Numerical: Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.
Solution: p(x)= x2 + 7x + 10= (x + 2)(x + 5)
p(x) = 0, when x= -2 or x= -5, therefore zeroes of p(x)= x2 + 7x + 10 are -2 & -5.
Let’s see if α + β = -b/a & α * β = c/a here a=1, b=7 & c =10
Sum of Zeroes = α + β = -2 + -5 = -7 -(i)
-b/a = -7/1 = -7 -(ii)
Comparing i & ii, α + β = -b/a
Product of Zeroes = α * β = -2 * -5 = 10 - (iii)
c/a = 10/1 = 10 - (iv)
Comparing iii & iv, α * β = c/a
Numerical: Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.
Solution: General form of quadratic polynomial is p(x) = ax2 + bx + c -(i)
α + β = -b/a = -3 or b = 3a -(ii)
& α * β = c/a = 2 or c = 2a -(iii)
Let a=k, then b=3k & c= 2k, using (ii) & (iii)
Thus the equation will be p(x) = kx2 + 3kx + 2k or p(x) = k(x2 + 3x + 2) where k is any real number.