Question

Find the cubic polynomial with the sum, sum of the products of its zeros taken two at a time and the product of the zeroes as 12, 39 and 28 respectively.​

Answer 1

ANSWER:

• Required polynomial x³-12x²+39x-28

GIVEN:

• Sum of zeros = 12
• Sum of the products of its zeros taken two at a time = 39.
• Product of zeros = 28

TO FIND:

• Cubic polynomial with the above information.

SOLUTION:

Cubic polynomial when Zeros are given:

= x³-(α+β+γ)x²+(αβ+βγ+αγ)x-αβγ. ....(i)

Here:

• α+β+γ = 12
• αβ+βγ+αγ = 39
• αβγ = 28

Putting the values in eq (i) we get;

= x³-(12)x²+(39)x-28

= x³-12x²+39x-28

Required polynomial = x³-12x²+39x-28

NOTE:

Some important formulas:

(a-b)³ = a³-b³-3ab(a-b)

a³+b³ = (a+b)(a²+b²-ab)

a³-b³ = (a-b)(a²+b²+ab)

(a+b)² = (a-b)²+4ab

(a-b)² = (a+b)²-4ab