Discovering Rational Roots for Simplifying Polynomials

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    Polynomial
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Discussion Overview

The discussion revolves around techniques for simplifying the polynomial $$x^3-9x^2+27x-27$$, with a focus on finding rational roots and factoring without using Horner's algorithm. Participants explore various methods and expressions related to polynomial simplification.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant inquires about the easiest way to simplify the given polynomial without using Horner's algorithm.
  • Another participant suggests rewriting the polynomial in a form resembling the expansion of $$(a+b)^3$$, specifically using $$x^3+3x^2(-3)+3x(-3)^2+(-3)^3$$.
  • There is a discussion about identifying $a$ and $b$ in the context of the cubic expansion, with some participants proposing that $a$ is $$x$$ and $b$$ is $$-3$$.
  • Participants express uncertainty about the next steps after identifying the values of $a$ and $b$, particularly regarding the factored form of the polynomial.
  • One participant confirms that the factored form is $$(x-3)^3$$ and identifies the root as $$3$$, while expressing frustration about their difficulty in solving such problems independently.
  • Another participant reassures that practice will help improve recognition of patterns in mathematics.
  • One participant suggests trying the rational roots theorem as an alternative method for finding roots.
  • A later reply provides a brief overview of the rational root theorem and lists potential rational roots for the polynomial.

Areas of Agreement / Disagreement

Participants generally agree on the identification of the factored form as $$(x-3)^3$$ and the root as $$3$$. However, there is no consensus on the best method for simplifying the polynomial, as multiple approaches are discussed.

Contextual Notes

Some participants express uncertainty about the application of the rational roots theorem and the process of polynomial division, indicating a need for further clarification on these methods.

Who May Find This Useful

This discussion may be useful for students or individuals seeking to understand polynomial simplification techniques, particularly those interested in rational roots and factoring methods.

theakdad
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I wonder what are the tecniques,or what is the easiest way to simplify given polynomial:

$$x^3-9x^2+27x-27$$

If possible,without Horners algorithm. Thank you!
 
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Perhaps if you write it as:

$$x^3+3x^2(-3)+3x(-3)^2+(-3)^3$$

Does this look like a familiar expansion?
 
MarkFL said:
Perhaps if you write it as:

$$x^3+3x^2(-3)+3x(-3)^2+(-3)^3$$

Does this look like a familiar expansion?

Yes MArk,but what is the next step?
What should i do with $$3^n$$ terms?
 
wishmaster said:
Yes MArk,but what is the next step?
What should i do with $$3^n$$ terms?

Consider that:

$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$

What are $a$ and $b$ in the case of the given expression?
 
MarkFL said:
Consider that:

$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$

What are $a$ and $b$ in the case of the given expression?

$a$ is $x$ and $b$ is $-3$ ?
 
wishmaster said:
$a$ is $x$ and $b$ is $-3$ ?

Yes, that's correct! :D

So, what is the factored form?
 
MarkFL said:
Yes, that's correct! :D

So, what is the factored form?

$$(x-3)^3$$ so the root of the polynomial is $3$.

I can't switch my brains to mathematical thinking...i get stucked by easy problems like this,and that is no good...
 
wishmaster said:
$$(x-3)^3$$ so the root of the polynomial is $3$.

I can't switch my brains to mathematical thinking...i get stucked by easy problems like this,and that is no good...

It comes with practice...you will find the more practice and experience you have, the more quickly you recognize patterns you have seen before. :D
 
MarkFL said:
It comes with practice...you will find the more practice and experience you have, the more quickly you recognize patterns you have seen before. :D

Yes,i think so...actually,i have no books,or something to learn,only online help. So here on the forum, and especially you MArk,are very helpfull for me...
 
  • #10
wishmaster said:
Yes,i think so...actually,i have no books,or something to learn,only online help. So here on the forum, and especially you MArk,are very helpfull for me...

We are glad to help here at MHB. :D

Also, I forgot to mention that your factored form is correct. (Yes)
 
  • #11
MarkFL said:
We are glad to help here at MHB. :D

Also, I forgot to mention that your factored form is correct. (Yes)

Thank you!

Yes,its correct,but I am not happy beacuse i didnt come alone to the solution...
 
  • #12
wishmaster said:
Thank you!

Yes,its correct,but I am not happy beacuse i didnt come alone to the solution...

Try another method then. Pretend you don't know the answer, and see if you can instead apply the rational roots theorem.
 
  • #13
MarkFL said:
Try another method then. Pretend you don't know the answer, and see if you can instead apply the rational roots theorem.

Wish i could know the other method...
 
  • #14
wishmaster said:
Wish i could know the other method...

Here is an article on it:

Rational root theorem - Wikipedia, the free encyclopedia

This theorem tells us that if the given polynomial has rational roots, it will come from the list:

$$\pm\left(1,3,9,27 \right)$$

So we let:

$$f(x)=x^3-9x^2+27x-27$$

and when we find a number $k$ from the list such that:

$$f(k)=0$$

then we know $x-k$ is a factor, and we may use polynomial or synthetic division to get:

$$f(x)=(x-k)P(x)$$

And then we see if we can then further factor $P(x)$.
 

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