Learn Exercise 2.4 with Free Lessons & Tips

Let the zeroes be α, β,γ.

Given sum of the zeros=α+β+γ = 2

Sum of the product of it's zeros taken two at a time = αβ+βγ+γα = -7

    Also, product of it's zeros=  αβγ= -14

The cubic polynomial is of the form

 x³ - (sum of the zeros) x² +(sum of the product of the zeros taken two at a time) - ( product of the zeros)=0

=> x³ - (α+β+γ)x² + (αβ+βγ+γα)x - (αβγ)=0

=> x³ -2x²-7x+14=0 which is the required polynomial.

(i) On comparing the given polynomial with the polynomial ax³ + bx² + cx + d, we obtain a = 2, b = 1, c = -5, d = 2 

Also, 

2+1-5+2 = 0 

Also, 

-16+4+10+2=0

Hence  are the zeroes. 

α=  β = 1 γ= -2

α+β+γ =  =  

αβ+βγ+γα = 

αβγ = 

Thus, the relationship between the zeroes and the coefficients is verified. 

(ii) On comparing the given polynomial with the polynomial ax^₃ + bx^₂ + cx + d, we obtain a = 1, b = -4, c = 5, d = -2. 

Hence 2, 1 and 1 are the zeroes. 

α= 2 β = 1 γ= 1

α+β+γ = 

αβ+βγ+γα = 

αβγ = 

Thus, the relationship between the zeroes and the coefficients is verified.

Let p(x) = x³ - 3x² + x + 1
The zeroes of the polynomial p(x) are given as a - b, a, a + b.
Comparing the given polynomial with dx³ + ex² + fx + g, we can observe that
d = 1, e = -3, f = 1, g = 1

Sum of the zeroes = α+β+γ = 

Product of the zeroes αβγ = 

Substituting the value of a from (i) and (ii), we get,

Hence a=1 and 

The 2 zeroes which are given are 

Therefore, x^₂ - 4x + 1 is a factor of the given polynomial.