1. Simplifying a long polynomial

This stems from a calculus problem, and I have no trouble with the calculus I only am having trouble in reducing the polynomial into a simplified form.

I am doing differentiations; the polynomial I end up with is

4[(4-3x^3)^3](-9x^2)(1-2x)^6+(4-3x^3)^4[6(1-2x)^5](-2)

and the reduced answer is the following but I don't know how to reduce it to that

12[(4-3x^3)^3][(1-2x)^5][(9x^3)(-3x^2)-4]

any help is appreciated.

2. or even if someone can link me to a possible tutorial that can explain. thank you

This stems from a calculus problem, and I have no trouble with the calculus I only am having trouble in reducing the polynomial into a simplified form.

I am doing differentiations; the polynomial I end up with is

4[(4-3x^3)^3](-9x^2)(1-2x)^6+(4-3x^3)^4[6(1-2x)^5](-2)

and the reduced answer is the following but I don't know how to reduce it to that

12[(4-3x^3)^3][(1-2x)^5][(9x^3)(-3x^2)-4]

any help is appreciated.
Hmm. So you have

$4(4-3x^{3})^{3}(-9x^{2})(1-2x)^{6}+(4-3x^{3})^{4}(6)(1-2x)^{5}(-2).$

Is that correct? If so, I would start by writing it as follows:

$4(4-3x^{3})^{3}(-9x^{2})(1-2x)^{6}-12(4-3x^{3})^{4}(1-2x)^{5}.$

Then, I would find the greatest common factor of these two terms. What is that?

4. yes that is correct

5. So what's the greatest common factor?

6. working on it now, I was trying to figure out how to use LaTex so my expressions were easy to read.

I would say 3?

7. Hmm. You're not thinking big enough. In expressions like this, the polynomials contribute to the greatest common factor as well. Let me write this out for you:

$\underbrace{4(4-3x^{3})^{3}(-9x^{2})(1-2x)^{6}}_{\text{This is one term,}}-\underbrace{12(4-3x^{3})^{4}(1-2x)^{5}}_{\text{and this is another.}}$

So, looking at ALL the factors that make up each term, including the expressions with x's in them, what is the greatest common factor? For example, what is the highest power of the factor

$4-3x^{3}$

that occurs in BOTH terms?

8. greatest common factors would be (4-3x^3)^4 and (1-2x)^6 and 3?

9. Closer, but still no cigar. Don't forget that there's a -9 multiplying the first term (though it's hidden in the middle there). Also, we're talking about the greatest common factor, not the least common multiple. When you compute the greatest common factor, you must find the largest power of factors that BOTH TERMS have. The first term does not a fourth power of (4-3x^3), nor does the second term have a sixth power of (1-2x). Basically, when you're computing the greatest common factor, what you take is the smallest exponent of factors in common. Make sense?

To make it more concrete, suppose I wanted to compute the greatest common factor of 36 and 100. First step: write each number in terms of prime factorizations. So, I get

36 = (2^2)(3^2)(5^0), and
100 = (2^2)(3^0)(5^2).

You'll see that I have inserted some fancy ways of writing 1 in there to illustrate that the principle extends to factors that are not in common.

So I look for the lowest exponents of common factors. That is, the greatest common factor is going to be

(2^2)(3^0)(5^0) = 4.

Do you see how I did that? That's exactly the sort of thing you need to do, only with polynomial terms instead of just numbers. The same rules apply.

So now, armed with this method, what do you get for the greatest common factor?

10. okay, that helps, I hope this is closer then (4, (4-3x^3)^3, (1-2x)^5)

11. Yes, much closer. But your number is off now! What's the greatest common factor of 12 and 36?

12. 2^2 and 3

13. I think you've basically got it, but you need to write it correctly (the way you present things is incredibly important - The Challenger blew up because the engineers who recognized the danger didn't present their results convincingly enough to their superiors!) Your answer should be one number. Recall that

12 = (3^1)(2^2), and
36 = (3^2)(2^2),

so the greatest common factor is what?

14. so it'd be 3^1 and 2^2?

15. You're presenting your answer wrong! The greatest common factor of two numbers is a single number. What is it? You're giving me two numbers. I don't know what that means (playing devil's advocate here), since the answer's supposed to be a single number.

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