Admittedly, I've touched a modicum bit in this regard at this slideshow (http://masteringolympiadmathematics.blogspot.com/2015/06/slideshow-9-creative-teaching.html), you could, if you want to refer it and answer the following quiz questions.

Question 1: What is the next square number after $(x-6)^4$?

$((x-6)^2)^2+1$

$((x-6+1)^2)^2$

$((x-6)^2+1)^2$

**Answer:**

If this is not immediately straightforward to you, I suggest to try out some simple cases so you would get a clear picture the pattern that dictates the relation.

Consider the following cases:

$x^2$ $(x+1)^2$

$x=1$ $1^2$ $2^2$

$x=2$ $2^2$ $3^2$

$x^2$ $(x+1)^2$

$x=1$ $1^2$ $2^2$

$x=2$ $2^2$ $3^2$

**$(6-x)^2$ $(6-x+1)^2$**

$x=1$ $5^2$ $6^2$

$x=2$ $4^2$ $5^2$

$x=1$ $5^2$ $6^2$

$x=2$ $4^2$ $5^2$

**$(x^3+6)^2$ $(x^3+6+1)^2$**

$x=1$ $7^2$ $8^2$

$x=2$ $14^2$ $15^2$

$x=1$ $7^2$ $8^2$

$x=2$ $14^2$ $15^2$

$(x^2)^2$ $(x^2+1)^2$

$x=1$ $1^2$ $2^2$

$x=2$ $4^2$ $5^2$

$(x^2)^2$ $(x^2+1)^2$

$x=1$ $1^2$ $2^2$

$x=2$ $4^2$ $5^2$

As you can see it by now, we have enough to start talking about the pattern...in order to get the next square number after a specific square, what we are need to do is by adding $1$ to the expression inside of the given square.

Note that we could rewrite $(x-6)^4$ as a square: [MATH]\color{black}\bbox[5px,orange]{((x-6)^2)}\color{black}^2[/MATH]

Therefore, the next square number after $(x-6)^4$ would be [MATH]\color{black}\bbox[5px,orange]{((x-6)^2\color{red}+1\color{black})}\color{black}^2[/MATH].

The answer is of course $((x-6)^2+1)^2$.

Question 2: What is/are the first thing that comes to your mind when you see $x^4-24 x^3+216 x^2-804 x+936$?

It has a repeated factor, perhaps?

We can factor it using the rational root test.

Binomial expansion.

**Answer:**

The first choice would be out of the question as the valid answer, because we would only "think and relate" of that after we have done some differentiation test. It should not be something that springs to our mind whenever we see any polynomial function that we suspect the polynomial has some or even one repeated root.

Yes, we can always factor the given polynomial using the rational root test, in fact, the constant $936$ that can be factored as $2^3\cdot 3^2\cdot 13$ has offered us much choices to figure out the possible root(s) for that polynomial.

Perhaps you are only familiar with the binomial cube where $(x-1)^3=x^3-3x^3+3x-1$, but, studies have shown that students don't get anything from going over too familiar stuffs, so, you must be very sensitive to something like

$(x-2)^3=x^3-6x^3+12x-8$

$(2x-1)^3=8x^3-12x^3+6x-1$

$(x-6)^3=x^3-18x^3+108x-216$

$(x-6)^4=x^4-24 x^3+216 x^2-864 x+1296$

If you could "see" it, you would realize at an instant that:

$\begin{align*}x^4-24 x^3+216 x^2-804 x+936&=(x^4-24 x^3+216 x^2-864 x+1296)+60x-360\\&=(x-6)^4+60(x-6)\\&=(x-6)((x-6)^3+60)\end{align*}$

Therefore, the correct answers to this question are the second and third option.

Question 3: What is the best equivalent representation of $x^4-24 x^3+216 x^2-804 x+936$ written as a square number?

$(x-6)\sqrt{((x-6)^3+60)^2}$

$(x^3-18 x^2+108 x-156)\sqrt{x-6}$

$((x-6)^2)^2+60(x-6)$

**Answer:**

From the above, we have already gotten

$\begin{align*}x^4-24 x^3+216 x^2-804 x+936&=(x-6)^4+60(x-6)\\&=(x-6)((x-6)^3+60)\end{align*}$

To rewrite it as a square, we would use the form $(x-6)^4+60(x-6)$:

$\begin{align*}x^4-24 x^3+216 x^2-804 x+936&=(x-6)^4+60(x-6)\\&=((x-6)^2)^2+60(x-6)\end{align*}$

When we got that, we know and could be sure that the main and key square term that is controlling the expression $x^4-24 x^3+216 x^2-804 x+936$ is [MATH]\color{yellow}\bbox[5px,green]{((x-6)^2)^2}[/MATH].

Question 4: If $x\ge 6$, what are the two squares of consecutive numbers that the number represented by $x^4-24 x^3+216 x^2-804 x+936$ lie between?

$(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2)^2+1$

$(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6+1)^2)^2$

$(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2+1)^2$

**Answer:**

Question 1 tells us, if we have [MATH]\color{yellow}\bbox[5px,green]{((x-6)^2)^2}[/MATH], then its next square term would be [MATH]\color{green}\bbox[5px,yellow]{((x-6)^2+1)^2}[/MATH].

Also note that $((x-6)^2)^2$ is smaller than $x^4-24 x^3+216 x^2-804 x+936=((x-6)^2)^2+60(x-6)$ for $x\ge 6$, we can conclude that $(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2+1)^2$ is the correct answer.

Question 5: Given that $x$ is an integer, and based on your answer in question 4, find the range of values of $x$ that satisfies the RHS of the inequality?

$x\gt 35$

$x\gt 36$

$x\gt 6$

**Answer:**

We obtained $(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2+1)^2$ in question 4, and we're now asked to solve for the range of $x$ that satisfies the RHS of the inequality.

In fact, what we're dealing now is $((x-6)^2)^2+60(x-6)\lt ((x-6)^2+1)^2$.

Solving it for $x$, we see that:

$(x-6)^4+60(x-6)\lt (x-6)^4+2(x-6)^2+1$

$\cancel{(x-6)^4}+60(x-6)\lt \cancel{(x-6)^4}+2(x-6)^2+1$

$60(x-6)\lt 2(x-6)^2+1$

$2x^2-84x+433\gt 0$

$x\lt \dfrac{42-\sqrt{898}}{2}\approx 6.0167$ or $x\gt \dfrac{42+\sqrt{898}}{2}\approx 35.9833$

Since we're informed that $x\ge 6$ and $x$ is an integer, so we obtain:

$x\gt 36$ and that is the answer to this question.

Question 6: Based on your answer in question 4 and 5, what can you say about the expression $x^4-24 x^3+216 x^2-804 x+936$, if it is meant to be a square?

$x^4-24 x^3+216 x^2-804 x+936$ can't be a square in a certain domain.

$x^4-24 x^3+216 x^2-804 x+936$ can't be a square for all real x..

$x^4-24 x^3+216 x^2-804 x+936$ is always a square.

**Answer:**

The answer we obtained from question 5 tells us $(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2+1)^2$ holds for $x\gt 36$.

We need to interpret what does it mean exactly by that. It has mathematical meaning but we need to also know what kind of information we could grab from that.

Spend as much time as possible to ponder about it, you would realize that $x^4-24 x^3+216 x^2-804 x+936$ cannot be a square at the interval where $(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2+1)^2$ is true, i.e. at $x\gt 36$.

So, if we want to force the quantity $x^4-24 x^3+216 x^2-804 x+936$ to be a square, we should look for the interval outside that range of $x$ that guarantees the inequality $(x-6)^4\lt x^4-24 x^3+216 x^2-804 x+936\lt ((x-6)^2+1)^2$ a true statement.

That is to say, $x^4-24 x^3+216 x^2-804 x+936$ can be a square at $x\le 35$.

You might not convince that what you have gotten so far is correct, but you can always ascertain it by trying to work out the problem numerically:

At $x=36$, $810,000\lt 811,800\lt 811, 801$

At $x=35$, $707,281\lt 709,021\lt 708,964$

At $x=7$, $1\lt 61\lt 4$

Bingo! We could now safely declare that $x^4-24 x^3+216 x^2-804 x+936$ can't be a square in a certain domain is the answer to this last question in this quiz!

Something that I want to remind you is that this quiz serves as a foundation to help guide in solving this year 2015 very challenging Olympiad Mathematics Problem that I will post it soon.

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