**QUADRATIC POLYNOMIAL**

Relationship between zeroes and coefficients

General form of Quadratic polynomial: ax^{2} + 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) = x^{2} – (α + β)x + αβ

= x^{2} – (–3)x + 2 = x^{2} + 3x + 2

Hence, required quadratic polynomial is x^{2} + 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) = x^{2} – (α + β)x + αβ

= x^{2} – (–1)x + (– 6) = x^{2} + x – 6

Hence, required quadratic polynomial is x^{2} + 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 x^{2} + 7x + 10, and verify the relationship between the

zeroes and the coefficients.

Q.6. Find the zeroes of the polynomial x^{2} – 3 and verify the relationship between the zeroes and the

coefficients.

Q.7. Find the zeroes of the quadratic polynomial 6x^{2} – 3 – 7x and verify the relationship between the

zeroes and the coefficients.

Q.8. Find the zeroes of the quadratic polynomial 3x^{2} – x – 4 and verify the relationship between the

zeroes and the coefficients.

Q.9. Find the zeroes of the polynomial x^{2}+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 4x^{2} – 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) = x^{2} – 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) = 2x^{2} – 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) = x^{2} – 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) = x^{2} – 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) = 2x^{2} + 5x + k such that α^{2}+β^{2}+αβ=21/4, find the value of k.