Question

if a,b,c are the roots of the equation:
${x}^{3} - 7 {x}^{2} + 12x - 11 = 0$
,Find the last four digit of
${a}^{21007} + {b}^{21007} + {c}^{21007}$
​

Answer 1

Answer:

Let the roots be a,3a,b.

The Sum of the roots is =4a+b=7 .. (i)

(a)(3a)+(a)(b)+(3a)(b)=4

⇒3a

2

+4ab=4 .. (ii)

The Product of roots is 3a

2

b=−12 .. (iii)

Substituting the value of b from (i) in (ii), we get ,

3a

2

+4a(7−4a)=4

⇒13a

2

−28a+4=0

⇒(a−2)(13a−2)=0

⇒a=2 or a=

13

2

For a=2 , b=−1 (By substituting in (i)). This satisfies equation (iii).

For a=

13

2

, b=

13

83

, this does not satisfy equation (iii).

Hence, the roots are (−1,2,6).