Let zeros or roots of a quadratic quadrilateral be a and b

Example: Form the quadratic polynomial whose zeroes or roots are 4 and 6.

Solution: **Sum of the zeroes** = 4 + 6 = 10**Product of the zeroes** = 4 × 6 = 24

Hence the polynomial formed by the given equation

= x^{2} – (sum of zeroes) x + Product of zeroes

= x^{2} – 10x + 24

Example: Form the quadratic polynomial whose zeroes are –3, 5.

Solution: Here, zeroes are – 3 and 5.**Sum of the zeroes** = – 3 + 5 = 2**Product of the zeroes** = (–3) × 5 = – 15

Hence the polynomial formed by

= x^{2} – (sum of zeroes) x + Product of zeroes

= x^{2} – 2x – 15

Example: Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively (i) , –1 (ii) √2, (iii) 0, √5

Solution: Let the polynomial be ax^{2} + bx + c and its zeroes be a and b.

(i) Here, α + β = ** **and α . β = – 1, therefore, the polynomial equation is formed by the formula

= x^{2} – (Sum of zeroes) x + Product of zeroes

= x^{2} – x – 1

The other polynomial are k.

If k = 4, then the polynomial is 4x^{2} – x – 4.

(ii) Here, α + β = √2 , α . β = . Thus the polynomial formed by

= x^{2} – (Sum of zeroes) x + Product of zeroes

= x^{2} – (√2) x + ** **

Other polynomial are k.

If k = 3, then the polynomial is 3x^{2} – 3√2x + 1

(iii) Here, α + β = 0 and α . β = √5, Thus the polynomial formed

= x^{2} – (Sum of zeroes) x + Product of zeroes

= x^{2} – (0) x + √5 = x^{2} + √5

Example: Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time, and product of its zeroes as 2, – 7 and –14, respectively.

Solution: Let the cubic polynomial be ax^{3} + bx^{2} + cx + d, divide the whole polynomial by a, we get

⇒ x^{3} + x^{2} + x + ....(1) and its zeroes are a, b and g, then

a + b + g = 2 = –

ab + bg + ag = – 7 =

abg = – 14 = –

Putting the values of , and , in equtaion number 1, we get the cubic polynomial equation is

x^{3} + (–2) x^{2} + (–7)x + 14

⇒ x^{3} – 2x^{2} – 7x + 14

Example: Find the cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and product of its zeroes as 0, –7 and –6 respectively.

Solution: Let the cubic polynomial be ax^{3} + bx^{2} + cx + d, then

⇒ x^{3} + x^{2} + x + ....(1) and its zeroes are a, b, g. Then

a + b + g = 0 = –

ab + bg + ag = – 7 =

abg = – 6 = –

Putting the values of , and ,, and in (1), we get

x^{3} – (0) x^{2} + (–7) x + (–6)

or x^{3} – 7x + 6

Example: If a and b are the zeroes of the polynomials ax^{2}+ bx + c then form the polynomial whose zeroes are and .

Solution: Since a and b are the zeroes of ax^{2} + bx + c

So a + b = , ab =

Sum of the zeroes = ** **+ =

= =

Product of the zeroes

= × ==

But required polynomial is given by the equation x^{2} – (sum of zeroes) x + Product of zeroes

⇒ x^{2} – x +

⇒ c

Example: If a and b are the zeroes of the polynomial x^{2}+ 4x + 3, form the polynomial whose zeroes are 1 + and 1 + .

Solution: Since a and b are the zeroes of the polynomial x^{2} + 4x + 3. Then, a + b = – 4, ab = 3

Sum of the zeroes

= 1 + ** **+ 1 + =

= = = =

Product of the zeroes

= (1 + )(1 + )= 1 + + + 1

= 2 + =

= ==

But required polynomial is given by the quadratic formula

x^{2} – (sum of zeroes) x + product of zeroes

or x^{2} – x + or k

or 3(if k = 3)

⇒ 3x^{2} – 16x + 16

SHOW CHAPTERS