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.

If you want to learn more, you can read:

link

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.

120 seems like the most sensible answer.

step-by-step explanation:

if there is only one topping on all caramel apples and there's 120 of them. then all apples should/will be covered in sprinkles. make sense?