QUADRATIC POLYNOMIAL
Relationship between zeroes and coefficients
General form of Quadratic polynomial: ax2 + bx + c, a ≠ 0
Sum of zeroes (α + β) = -b/a; Product of zeroes (αβ) = c/a
Question: Find a quadratic polynomial, the sum and product of whose zeroes are – 3 and 2, respectively.
Solution: Here, α + β = – 3 and αβ = 2
We know that quadratic polynomial is given by p(x) = x2 – (α + β)x + αβ
= x2 – (–3)x + 2 = x2 + 3x + 2
Hence, required quadratic polynomial is x2 + 3x + 2
Question: Find a quadratic polynomial, whose zeroes are – 3 and 2.
Solution: Here, α = – 3 and β = 2.
Now, α + β = – 3 + 2 = – 1 and αβ = (– 3)(2) = – 6
We know that quadratic polynomial is given by p(x) = x2 – (α + β)x + αβ
= x2 – (–1)x + (– 6) = x2 + x – 6
Hence, required quadratic polynomial is x2 + x – 6
Q.1. Find a quadratic polynomial, the sum and product of whose zeroes are – 5 and 3, respectively.
Q.2. Find a quadratic polynomial, whose zeroes are – 4 and 1, respectively.
Q.3. Find a quadratic polynomial, the sum and product of whose zeroes are √2 and -3/2, respectively. Also find its zeroes.
Q.4. For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
(i) -8/3, 4/3
(ii) 21/8, 5/16
(iii) -2√3, -9
(iv) -3/2√5, -1/2
Q.5. Find the zeroes of the quadratic polynomial x2 + 7x + 10, and verify the relationship between the
zeroes and the coefficients.
Q.6. Find the zeroes of the polynomial x2 – 3 and verify the relationship between the zeroes and the
coefficients.
Q.7. Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the
zeroes and the coefficients.
Q.8. Find the zeroes of the quadratic polynomial 3x2 – x – 4 and verify the relationship between the
zeroes and the coefficients.
Q.9. Find the zeroes of the polynomial x2+x/6-2, and verify the relation between the coefficients and the zeroes of the polynomial.
Q.10. Find the zeroes of the quadratic polynomial 4x2 – 4x + 1 and verify the relationship between the
zeroes and the coefficients.
Q.11. If α and β are the zeroes of the quadratic polynomial f(x) = x2 – 3x – 2, then find a quadratic polynomial whose zeroes are 1/(2α+β) and 1/(2β+ α).
Q.12. If α and β are the zeroes of the quadratic polynomial f(x) = 2x2 – 5x + 7, then find a quadratic
polynomial whose zeroes are 2 α + 3 β and 2 β + 3 α .
Q.13. If α and β are the zeroes of the quadratic polynomial f(x) = x2 – 1, then find a quadratic
polynomial whose zeroes are 2α/β and 2β/α.
Q.14. If α and β are the zeroes of the quadratic polynomial f(x) = x2 – 2x + 3, then find a quadratic
polynomial whose zeroes are α+2 and β+2.
Q.15. If α and β are the zeroes of the quadratic polynomial f(x) = 2x2 + 5x + k such that α2+β2+αβ=21/4, find the value of k.