The sum of the first n perfect cubes is given by the formula S=(1)/(4)(n^4+2 n^3+n^2) . Use the

step 1 So in this problem, we're looking for two things. The first part, part A, we're looking for the

The sum of the first n perfect cubes is given by the formula...

The sum of the first $n$ perfect cubes is given by the formula $S=\frac{1}{4}\left(n^{4}+2 n^{3}+n^{2}\right) .$ Use the remainder theorem on $S$ to find the sum of (a) the first three perfect cubes (divide by $n-3$ ) and (b) the first five perfect cubes (divide by $n-5$ ). Check results by adding the perfect cubes manually. To avoid working with fractions you can initially ignore the $\frac{1}{4}\left(\text { use } n^{4}+2 n^{3}+n^{2}+0 n+0\right),$ as long as you divide the remainder by 4.

Key Concepts

Remainder Theorem

The remainder theorem states that when a polynomial f(x) is divided by a linear factor (x - c), the remainder is equal to the value of the polynomial at x = c, i.e., f(c). This is a powerful tool for quickly determining remainders without performing full-fledged polynomial division.

Polynomial Evaluation

Polynomial evaluation involves substituting a particular value into a polynomial expression to compute its result. This approach is fundamental when applying the remainder theorem, as it allows one to directly compute the remainder by calculating f(c).

Closed-Form Summation Formulas

Closed-form summation formulas offer a precise method to calculate the sum of a sequence without needing to add each individual term manually. These formulas, which often encapsulate sequences of power sums, streamline calculations and provide insights into underlying patterns within numerical series.

Verification through Direct Computation

Verification through direct computation entails manually adding the individual terms of a series to confirm the correctness of results obtained through formulas or theorems. This method serves as an essential check to ensure that derived results are accurate and reliable.

Transcript

00:01 So in this problem, we're looking for two things.

00:03 The first part, part a, we're looking for the sum of the first three perfect cubes.

00:09 And it gives us a clue to divide by n minus three.

00:12 And then for b, it's looking for the sum of the first five perfect cubes.

00:17 And it also gives us a clue to divide by an n minus five.

00:22 So we're going to go ahead and set this up.

00:26 The first thing we're going to do is, you know, working with fractions can kind of get a little icky.

00:30 We're going to take that one fourth and we're just going to kind of put it off to the side and we're going to work with n to the fourth plus two n cubed plus n squared now once we get our answer we're going to divide it by four and that's when we'll bring that floor back in so we're going to go ahead we'll do a first so we're going to go ahead and set up synthetic division so we've got three and we got one two and one now we also need our placeholders.

01:07 So we've got two zeros.

01:09 So let's go ahead and solve this.

01:12 We got one times three is three.

01:14 Two plus three is five.

01:16 Five times three is 15.

01:19 One plus 15 is 16.

01:22 16 times three is 48.

01:27 So this is going to be 48.

01:29 And then 48 times three is 144.

01:32 Now remember, this is not our answer because we have this one -fourth.

01:36 So we need to take that 144 and divide by four.

01:44 So this goes in three times.