Prove that if r is any rational number, then 3r2 − 2r + 4 is rational. The following properties may be used in your proof. Property 1: Every integer is a rational number. Property 2: The sum of any two rational numbers is rational. Property 3: The product of any two rational numbers is rational. Note: Property 1 is Theorem 4.3.1, Property 2 is Theorem 4.3.2, and Property 3 is Exercise 15 in Section 4.3. Using these properties, choose explanations for each step in the given proof.

We want to prove that if r is rational, then:

 Is also rational.

The proof is below:

Remember that a rational number is any number that can be written as a quotient of two integer numbers.

Then we have:

Where a and b are integers.

If we replace that in our equation, we get:

Note that if a is an integer, then 3*a^2 is also an integer.

If b is an integer, then b^2 is also an integer.

Now we can perform the sum of these fractions to get:

And because integers are closed under the addition, then the numerator of that fraction is also an integer, then we have that the given expression can be written as a quotient of two integers, thus, it is a rational number.

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Given:

Expression is

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,

It can be written as

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.

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step-by-step explanation:

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