Question

If LCM of polynomials (y-3)^a (2y+1)^b (y+13)^7 and (y-3)^4 (2y+1)^9 (y.+13)^c is(y-3)^6 (2y+1)^10(y+13)^7, then the least value of a + b + c is?

Answer 1

polynomial 1,P1 =(y-3)^a (2y+1)^b (y+13)^7

polynomial 2,P2 =(y-3)^4 (2y+1)^9 (y.+13)^c

LCM of P1 and P2 ,P3 =(y-3)^6 (2y+1)^10(y+13)^7,


WE can take term by term

For P1 (y-3)^4 ,P2 =(y-3)^a and P3 =(y-3)^6

For LCm of (y-3)^4 and (y-3)^a =(y-3)^4+a

So Given 4+a=6

Then least value of a =6-4 =2


For P1 =(2y+1)^b ,P2 = (2y+1)^9 and P3 =(2y+1)^10

For LCm of (2y+1)^b and (2y+1)^9 =(2y+1)^9+b

So given 9+b=10

Then least value of b =10-9 =1


For P1 =(y+13)^7 ,P2 =(y.+13)^c and P3 =(y+13)^7

For LCm of (y+13)^7 and (y.+13)^c =(y+13)^7+c

So given 7+c=7

Then least value of b =7-7 =0


THere fore least value of a+b+c =2+1+0=3