Question

Find a quadratic polynomial whose sum and product of the zeroes are 4 and 1 respectively. Also find the zeroes?

Answer 1

α+β=4
αβ=1                                                           
=k(x²-(α+β)x+αβ)
 =k(x²-4x+1)
polynomial is x²-4x+1 
k=1

Answer 2

Given,

• Sum of the zeroes = 4
• Product of the zeroes = 1

To find,

We have to find the

• quadratic polynomial
• the zeroes of the polynomial

Solution,

We can simply find the quadratic polynomial by using the following formula:

         x² -(sum of the zeroes)x + (product of the zeroes)  (*)

Sum of the zeroes = 4

Product of the zeroes = 1

Using (*), we get

        x² - 4x + 1

which is the required quadratic polynomial.

Now, for the zeroes of the polynomial, let us factorize the polynomial by using the quadratic formula:

             x = -b±√D/2a   (**)

             D = b²-4ac

where b = -4, a = 1, c = 1

            D = (-4)²-4(1)(1)

            D = 16-4

            D = 12

Using (**), we get

             x = -(-4) ± √12/2(1)

             x = 4 ± 2√3/2

            x = 4+2√3/2,   x = 4-2√3/2

Hence, the quadratic polynomial whose sum and product of the zeroes are 4 and 1 respectively is x² - 4x + 1, and the zeroes of the polynomial are 4+2√3/2 and 4-2√3/2.