1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# If the rational number pq, q≠0 (where p and q are relatively prime) is a root of the equation, anxn+an−1xn−1+...+a1x+a0=0,where a0, a1, a2,...,an are integers and an≠0, then show that p is a divisor of a0 and q that of an.

Open in App
Solution

## Given pq is a root.⇒ an(pq)n+an−1(pq)n−1+...+a1(pq)+a0=0⇒ anpn+an−1qpn−1+...+a1qn−1p+a0qn=0 (i)⇒ an−1pn−1+an−2pn−2q+...+a1qn−2p+a0qn−1=−anpnq (ii)⇒ Here, a0, a1, ..., an−2, an−1, p,q∈Integers.⇒ LHS is an integer, so RHS is also an integer.ie, −anpnq is an integer, where p and q are relatively prime to each other.Thus, q must divide an.Again, anpn+an−1pn−1q+...+a1qn−1p=a0qn⇒ anpn−1+an−1qpn−2+...+a1qn−1=a0qnp (iii)As from above; a0qnp∈Integer⇒p is divisor of a0 (as p and q are relatively prime)thus if the rational number pq is root ofanxn+an−1xn−1+...+a1x+a0=0,then p divides a0 and q divides an.

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Integers
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program