MHB Discovering Rational Roots for Simplifying Polynomials

[theakdad]

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!

 

[MarkFL]

Perhaps if you write it as:

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

Does this look like a familiar expansion?

 

[theakdad]

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?

 

[MarkFL]

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?

 

[theakdad]

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 $-3$ ?

 

[MarkFL]

$a$ is $x$ and $b$ is $-3$ ?

Yes, that's correct! :D

So, what is the factored form?

 

[theakdad]

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...

 

[MarkFL]

$$(x-3)^3$$ 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

 

[theakdad]

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...

 

[MarkFL]

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)

 

[theakdad]

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...

 

[MarkFL]

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.

 

[theakdad]

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...

 

[MarkFL]

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 $P(x)$.