Proofs

Hi

Can anyone help me prove if a and b are rational numbers, then a(b^2) - 5 is a rational number.

If a and b are rational, then they can be written as the quotient of 2 integer numbers, for instance:

a = p/q
b = v/w

where p,q,v,w belong to Z (the set of integer numbers)

So, then

ab² - 5 = (p/q)(v/w)² - 5 = (pv² - 5qw²) / (qw²)

which in its turn is the quotient of integer numbers since only sums, substractions and products are involved in both the numerator and the denominator.

Not to be picky would it still be rational if q=w=0? or q=0 or w=0?

Neither q nor w can be zero, because a and b are given to be rational.

If a=0 and b=0 then

ab² - 5 = - 5

which is rational.

And 0 itself is rational. It can be expressed as a quotient:

0 = 0 / A

where A is any integer number different from 0.