Given:
Expression is
![3r^2-2r+4](https://tex.z-dn.net/?f=3r%5E2-2r%2B4)
To prove:
If r is any rational number, then
is rational.
Step-by-step explanation:
Property 1: Every integer is a rational number. It is Theorem 4.3.1.
Property 2: The sum of any two rational numbers is rational. It is Theorem 4.3.2.
Property 3: The product of any two rational numbers is rational. It is Exercise 15 in Section 4.3.
Let r be any rational number.
We have,
![3r^2-2r+4](https://tex.z-dn.net/?f=3r%5E2-2r%2B4)
It can be written as
![3(r\times r)-2r+4](https://tex.z-dn.net/?f=3%28r%5Ctimes%20r%29-2r%2B4)
Now,
3, -2 and 4 are rational numbers by property 1.
is rational by Property 3.
are rational by Property 3.
is rational by property 2.
So,
is rational.
Hence proved.