I have been told that ChatGPT is evolving and learning in real time. It's understandable that as more people use artificial intelligence it grows and becomes more complex just like a human brain. So, I decided to feed ChatGPT some mathematical problems to see what it is capable of handling. I was impressed with AI's ability to solve complex algebra problems instantly, so I decided to ask it to factor some polynomials next. After a few easy problems I fed it a cubic which it handled well. After more exercises, I was confident I had found a limit when it stumbled with a quartic.

#### Here is what I asked ChatGPT to solve:

**TC: "factor x^4-4x^3+4x^2-36x-45 to its simplest form"**

**ChatGPT Response:**

**"To factor the polynomial x^4 - 4x^3 + 4x^2 - 36x - 45 into its simplest form, we can start by trying to find the roots of the polynomial. Once we find the roots, we can use them to factor the polynomial.**
**First, let's check if there are any rational roots using the Rational Root Theorem. The rational root theorem states that if there are any rational roots (in the form p/q), p is a factor of the constant term, and q is a factor of the leading coefficient. In this case, the leading coefficient is 1, and the constant term is -45. So the possible rational roots are factors of 45: ±1, ±3, ±5, ±9, ±15, ±45.**
**By testing these values, you'll find that x = -3 is a root of the polynomial:**
**x^4 - 4x^3 + 4x^2 - 36x - 45 = 0 (-3)^4 - 4(-3)^3 + 4(-3)^2 - 36(-3) - 45 = 81 + 108 + 36 + 108 - 45 = 288 - 45 = 243 = 0**
**Now, we can factor the polynomial using synthetic division or polynomial long division:**
**(x^4 - 4x^3 + 4x^2 - 36x - 45) / (x + 3) = x^3 - 7x^2 + 25x + 15**
**Now, we have factored the polynomial as:**
**x^4 - 4x^3 + 4x^2 - 36x - 45 = (x + 3)(x^3 - 7x^2 + 25x + 15)**
**Next, let's factor the cubic polynomial x^3 - 7x^2 + 25x + 15. You can use various methods, such as grouping or the Rational Root Theorem, to find its roots and then factor it. After finding the roots, you can express the polynomial as a product of linear factors.**
**The factorization of the polynomial x^3 - 7x^2 + 25x + 15 is a bit more involved, and it may not have simple linear factors. You might want to use a numerical method or a computer algebra system to find the roots and factor it further if needed.**

Although it did not start with grouping, I was impressed how it understood and decided to use the Rational Root Theorem to solve this problem. However, it made what looks to me like a "human error" in the middle of the problem. A simple addition/subtraction error. Strange... I have highlighted the mistake in yellow above.

My first thought was that I had found a limit to ChatGPT's intelligence or problem-solving ability. And that beyond cubics, AI was not capable of factoring more complex polynomials.....But then I decided to see what would happen if I simply told ChatGPT, it had made a mistake.

#### Here what happened next:

**TC: "-3 is not a root, you have made a math error"**

**ChatGPT Response**:

" **I apologize for the mistake. Let's reevaluate the polynomial and find its roots correctly.**
**The polynomial is x^4 - 4x^3 + 4x^2 - 36x - 45. To find its roots, we can use the Rational Root Theorem to check for rational roots. The rational root theorem suggests that the rational roots must be among the factors of the constant term (-45) divided by the factors of the leading coefficient (1). The possible rational roots are ±1, ±3, ±5, ±9, ±15, and ±45.**
**Let's try these values one by one:**
**For x = 1: f(1) = 1^4 - 4***1^3 + 4***1^2 - 36*1 - 45 = 1 - 4 + 4 - 36 - 45 = -80**
**For x = -1: f(-1) = (-1)^4 - 4*(-1)^3 + 4*(-1)^2 - 36*(-1) - 45 = 1 + 4 + 4 + 36 - 45 = 0**
**So, x = -1 is a root of the polynomial.**
**Now, we can use synthetic division to divide the polynomial by (x + 1) to find the other factors:**
**(x^4 - 4x^3 + 4x^2 - 36x - 45) / (x + 1)**
**Performing synthetic division, we get:**
**x^3 - 5x^2 + 9x + 45**
**Now, we can factor the cubic polynomial x^3 - 5x^2 + 9x + 45. This can be factored as:**
**(x + 1)(x^3 - 5x^2 + 9x + 45) = (x + 1)(x^2(x - 5) + 9(x - 5)) = (x + 1)(x - 5)(x^2 + 9)**
**So, the fully factored form of the polynomial x^4 - 4x^3 + 4x^2 - 36x - 45 is:**
**(x + 1)(x - 5)(x^2 + 9)"**

As the sum of squares cannot be factored with real numbers, this is the correct solution.

ChatGPT accepted my critique. After a polite apology, it went to work again applying the Rational Root Theorem to compile rational roots. This time after evaluating rational roots, it correctly identified x = -1 as a root. After applying synthetic division and then grouping, it found x = 5 as a root and determined the fully factored form.

This was quite surprising to me. I have to admit, I still feel AI (ex. ChatGPT) is much like a calculator. If you plug in a question, you will get one answer. If you ask a calculator the same question twice, you will simply get the same answer twice. This is not the case with AI. Over time AI evolves, its knowledge expands, and it becomes more complex. It learns. It's a little unnerving to see a computer make a mistake and then observe it demonstrate (programmed) humility and provide an alternate solution. Let alone a correct solution.

**TC: "wow, I am impressed. thank you"**

**ChatGPT Response: **

** "You're very welcome! I'm glad I could help, and I appreciate your understanding. If you have any more questions or need further assistance with anything else, please feel free to ask. I'm here to help!"**

This exercise illustrates the caution we must use when using AI and not just casually accepting its answers or outputs. But to me it was more. An intimate moment between man and machine where I watched it work, make a mistake, try again, learn, and succeed. Like a person.....

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