The Golden Polynomial Puzzle

追加: 2026-05-30

[00:00:01]Hello everyone. In this video, we're going to be evaluating a polomial expression. We're given that x^2 - x is equal to 1. And we're going to evaluate x^ 5th power - 5x.

[00:00:15]We're going to be doing this in two different ways. And let's start with the first method.

[00:00:24]So for my first method, I'm going to go ahead and solve this quadratic equation.

[00:00:28]My goal is to find the x values that satisfy this equation and then substitute them into the second expression. Okay?

[00:00:40]Because we're looking for x values that satisfy the first one. And then for those x values, we need to evaluate the second expression. Okay? So let's go ahead and put everything on the same side.

[00:00:54]You'll probably recognize this um equation. It says famous one and if you write the solutions in a method you like I like the quadratic formula b + - the square<unk> of b ^ 2 which is 1 - 4 a c that's a + 4 / 2 that gives you 1 + minus<unk> 5 / 2 do you recognize these numbers hopefully now which value should we use?

[00:01:26]That is the million-dollar question, right? And I'd like to say I want to use the one with a plus sign because it looks more positive. The other one is definitely a negative number, isn't it?

[00:01:37]So, let's go ahead and evaluate this.

[00:01:39]So, I do need x to the 5th minus 5x, right? So, to be able to find it, obviously I know what x is. So, I do need to evaluate x to the fifth power.

[00:01:51]Now, I have different options. I can take the x raise it to the fifth power directly and then on the right hand side here we must use the binomial theorem or I can try raising it to the second power by squaring and then squaring it again which gives me the fourth power and then just multiply by x. So let's go ahead and use the second one. I'll take uh both sides to the second power. And when you take the right hand side, you're going to get a 2 b 2 a b. Remember the formula 1 + 5 + 2<unk> 5 / 4. This is 6 + 2<unk> 5 / 4. And from here x^2 = 3 +<unk> 5 / 2. Awesome. So, this is x squ. What I need to do now is square that one more time. Maybe not frame it like that. Let's go ahead and square both sides.

[00:02:57]And that'll give us x^ 4th power = 9 + 5 + 6<unk> 5. Again, by using the same formulaide by 4, this is 14 + 6<unk> 5 / 4. We can divide everything by two again. 7 + 3<unk> 5 / 2. So now we got x to the 4th power and we already know what x is. So x to the 5th is basically x 4 * x. So it is 7 + 3<unk> 5 over 2 * x which is 1 +<unk> 5 / 2. Right? Now to simplify this, we're going to go ahead and multiply the numerators. That's going to give us 7. And then 3 * 5 is 15. 7 + 15 is 22. And then you do the foil, you know, 7<unk> 5 and 3<unk> 5.

[00:03:52]That'll give us 10<unk> 5. And divide by 4 again. That is going to be 11 + 5<unk> 5 / 2, not four obviously. Okay. So we got x to the fifth power. The next step would be subtracting 5 * x. So we need x 5 - 5 x which is going to be 11 + 5<unk> 5 / 2 - 5 * what is x? 1 +<unk> 5 / 2.

[00:04:26]They have a common denominator. So I can go ahead and just subtract those. Of course, first you need to distribute.

[00:04:32]That gives you 5 + 5<unk> 5. So the numerator becomes 11 + 5<unk> 5 - 5 - 5<unk> 5 all over 2 5<unk> 5 miraculously mathematically cancels out 11 - 5 = 6 6 / 2 = 3 taa we got a numerical value of course that was the goal right now what would happen and this is what I want you to think about what would happen if you instead use x = 1 -<unk> 5 /2 instead of this. Right? I want you to go ahead and explore that and see if you can arrive at the same value. Okay? Now, let's go ahead and take a look at the second method. The second method definitely obviously uses a different approach and that's basically the purpose of this video because I want to show you some shortcuts. either you're preparing for a math contest or not, doesn't matter. But this is kind of to give you some ideas, uh, quick ways of problem solving and hopefully some elegant methods. Let me know what you think. I also have another channel that focuses on complex numbers.

[00:05:45]If you're interested, even if you're not interested, I'm sure you'll be interested. Check out a plus bi. All right. Now, the second method uses a polomial approach. But here's what we're going to do. We're going to evaluate powers of x without finding the exact value. Okay. So for that we'll do the following. From this equation we're going to isolate the x with the highest power which is x^2. So we can now write x^2 = x + one. This is kind of like linearization of any power of x. Make sense? So we're expressing it as a linear function. In other words, from here obviously we can come up with so many things. For example, you can get x to the 4th by squaring both sides. Since x^2 is equal to x + 1, if I square both sides just like before, but this time notice that you're not dealing with anything numerical. This gives you x 4th = x^2 + 2x + 1. And you always have to remember this. x^2 is equal to this. Okay? x + 1.

[00:06:55]So anytime anywhere you see this you can replace it with x + 1 and then of course it's followed by 2x + 1 which means x 4 x to the 4th was also linearized as 3x + 2. Okay that is x 4th. Now the next thing is x to the 5th. To be able to do that we're going to take x 4th and multiply by x. Does that look familiar?

[00:07:20]And x 4th can be replaced with 3x + 2.

[00:07:24]Multiply by x. Use the distributive property. 3x^2 + 2x. And then this is x to the 5th.

[00:07:32]Remember that. And now you're going to do the magical touch or mathematical touch. Maybe you'll replace x^2 with x + one. Again, again, notice that we're not finding the value of x here. We keep using the same thing over and over. And of course, this linearizes x to the fifth power. And that gives us 5x + 3.

[00:07:51]So what is the good thing about it?

[00:07:53]Right? Well, you already know x to the 2 power - x is 1 or x^2 is equal to x + one. And but wait a minute, how do you use this information? Okay, let's go back to the basics, right? What was the question again? Well, we were given this and we're supposed to evaluate this one, right? Okay, now take a look. We're going to do a super duper mathematical uh touch here because this is really going to solve the problem. Like boom, that's it. Right now, if you go ahead and subtract 5x from both sides, what do you get? x^ 5th - 5x is equal to 3. So, you're kind of solving the problem without solving the problem, whatever that means. And this brings us to the end of this video. Well, thanks for watching. Hope you enjoyed it. Please let me know. Don't forget to comment, like, and subscribe. I'll see you next time with another video. Until then, be safe, take care. Don't forget to check out the shorts channel. I have a shorts channel, too. A+BI and bye-bye.