34 points nita finds that the expression (a+1)^2 is sometimes, but not always, greater in value than (a+1)^3 when she evaluates both expressions for the same value of a. how do you explain this finding?

Consider the two expressions, (a + 1)² and (a + 1)³.

No matter what the number is, a² ≥ 0. Even if you square a negative number, you still get a positive number. We include the "or equal to" since 0² = 0. Since a² ≥ 0 is always true, then we can manipulate this inequality.

a² ≥ 0

a² + 2a + 1 ≥ 0 + 2a +1 by adding 2a +1 to both sides

(a + 1)² ≥ 2a + 1 as a² + 2a + 1 factors into (a +1)²

(a + 1)² ≥ 1 If it's more than 2a +1, it has to be more than 1

(a + 1)² ≥ 0 If it's more than one, it's more than zero since 1 > 0.

So we can conclude that (a + 1) ≥ 0 is always true.

Now let's look at (a + 1)³. We know that from before (a + 1)² ≥ 0. A tempting thing to say is that if you multiply by both sides by (a + 1) then both will be more than zero. Doing so isn't right.

What we instead must do is use cases. Either a is positive, negative, or zero.

Case 1: a = 0

When a = 0, (a + 1)³ = 1³ = 1 and 1 > 0. Thus (a + 1)³ ≥ 0,

Case 2: a > 0

When a > 0, a² > 0 from multiplying both sides by a. Do it a second time and a³ > 0. Then, if we add terms to both sides like in the a² example, we have this:

a³ > 0

a³ + 3a² > 3a² by adding 3a² to both sides

a³ + 3a² + 3a > 3a² + 3a by adding 3a to both sides

a³ + 3a² + 3a + 1 > 3a² + 3a + 1 by adding 1 to both sides

(a + 1)³ > 3a² + 3a + 1 by factoring the left side

(a + 1)³ > 1 Since a > 0 by assumption then 3a > 0 and 3a² > 0 and their sum is more than zero too

(a + 1)³ > 0 Since 1 > 0

Case 3: a < 0

Since a < 0 then a² > 0 (minus times minus is plus), but a³ < 0 through a similar multiplication.

Let b = a + 1. If a < 0 then a + 1 < 1 by adding 1 to both sides and b < 1 by back substitution. So b³ < 1³ by cubing both sides and b³ < 1 since 1³ =1.

b³ < 1

(a+1)³ < 1 We chose a + 1 = b.

When a < 0, we can conclude that (a + 1)³ < 1. When a ≥ 0, then (a +1)³ ≥ 0.

However for all a, (a +1)² ≥ 0. Thus, we have our sometimes truth. That when we choose a to be negative we have that the (a + 1)² and (a + 1)³ are of opposite signs.

Consider the two expressions, (a + 1)² and (a + 1)³.No matter what the number is, a² ≥ 0. Even if you square a negative number, you still get a positive number. We include the "or equal to" since 0² = 0. Since a² ≥ 0 is always true, then we can manipulate this inequality.a² ≥ 0a² + 2a + 1 ≥ 0 + 2a +1 by adding 2a +1 to both sides(a + 1)² ≥ 2a + 1 as a² + 2a + 1 factors into (a +1)²(a + 1)² ≥ 1 If it's more than 2a +1, it has to be more than 1(a + 1)² ≥ 0 If it's more than one, it's more than zero since 1 > 0.So we can conclude that (a + 1) ≥ 0 is always true.Now let's look at (a + 1)³. We know that from before (a + 1)² ≥ 0. A tempting thing to say is that if you multiply by both sides by (a + 1) then both will be more than zero. Doing so isn't right.What we instead must do is use cases. Either a is positive, negative, or zero.Case 1: a = 0When a = 0, (a + 1)³ = 1³ = 1 and 1 > 0. Thus (a + 1)³ ≥ 0,Case 2: a > 0When a > 0, a² > 0 from multiplying both sides by a. Do it a second time and a³ > 0. Then, if we add terms to both sides like in the a² example, we have this:a³ > 0a³ + 3a² > 3a² by adding 3a² to both sidesa³ + 3a² + 3a > 3a² + 3a by adding 3a to both sidesa³ + 3a² + 3a + 1 > 3a² + 3a + 1 by adding 1 to both sides(a + 1)³ > 3a² + 3a + 1 by factoring the left side(a + 1)³ > 1 Since a > 0 by assumption then 3a > 0 and 3a² > 0 and their sum is more than zero too(a + 1)³ > 0 Since 1 > 0Case 3: a < 0Since a < 0 then a² > 0 (minus times minus is plus), but a³ < 0 through a similar multiplication.Let b = a + 1. If a < 0 then a + 1 < 1 by adding 1 to both sides and b < 1 by back substitution. So b³ < 1³ by cubing both sides and b³ < 1 since 1³ =1.b³ < 1(a+1)³ < 1 We chose a + 1 = b.When a < 0, we can conclude that (a + 1)³ < 1. When a ≥ 0, then (a +1)³ ≥ 0.However for all a, (a +1)² ≥ 0. Thus, we have our sometimes truth. That when we choose a to be negative we have that the (a + 1)² and (a + 1)³ are of opposite signs.

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