1. ## points of intersection

i am not sure how to find the points of intersection of these 2 functions algebraically.

$\displaystyle y=x^{1/3}$ and $\displaystyle y=x^2+x-1$

i would normally do this

$\displaystyle x^{1/3}=x^2+x-1$ and solve for x. but in this case the cube is making this a nasty approach.

one try got me to here $\displaystyle 0=x^{5/3}+x^{2/3}-2$

but this seems like i have wondered into the weeds.

2. ## Re: points of intersection

Hi skoker!

It seems to me you have the right approach.
Raise both sides of your equation to the 3rd power and you get a 6th order polynomial.
It has 2 obvious roots which are the only real roots.

3. ## Re: points of intersection

I would go with your first approach, giving:

$\displaystyle x^2+x-x^{\small{\frac{1}{3}}}-1=0$

Let $\displaystyle u=x^{\small{\frac{1}{3}}}$ and write:

$\displaystyle u^6+u^3-u-1=0$

$\displaystyle u^3\left(u^3+1\right)-(u+1)=0$

$\displaystyle u^3(u+1)\left(u^2-u+1\right)-(u+1)=0$

$\displaystyle (u+1)\left(u^3\left(u^2-u+1\right)-1\right)=0$

$\displaystyle (u+1)\left(u^4\left(u-1\right)+u^3-1\right)=0$

$\displaystyle (u+1)\left(u^4\left(u-1\right)+(u-1)\left(u^2+u+1\right)\right)=0$

$\displaystyle (u+1)(u-1)\left(u^4+u^2+u+1\right)=0$

Thus, we get 2 points of intersection:

(-1,-1), (1,1)

4. ## Re: points of intersection

i guess it is the only way to raise the cube.

@MarkFL2

this substitution you did, what is it called?

if you use the normal algebra and raise both sides you get

$\displaystyle x^6+3x^5-5x^3+2x-1$ then you start the factor.

but what you did there looks like the U-substitution from calculus for finding the anti-derivative of a composite function.

where in the algebra is this type of substitution explained? maybe for parametric equations?

5. ## Re: points of intersection

Originally Posted by skoker
i guess it is the only way to raise the cube.

@MarkFL2

this substitution you did, what is it called?

if you use the normal algebra and raise both sides you get

$\displaystyle x^6+3x^5-5x^3+2x-1$ then you start the factor.

but what you did there looks like the U-substitution from calculus for finding the anti-derivative of a composite function.

Rational Zeros of Polynomials

6. ## Re: points of intersection

It is called "substitution".

If you have an equation with a difficult sub expression, you give the sub expression a name, say "u", and replace it.
Then you solve the equation, and when you're done, you back substitute "u".

Actually, substitution in integrals is a more advanced version of regular substitution.

7. ## Re: points of intersection

thanks for the info...

@ILikeSerena

i will go back and look at the substitution a bit. it seems in the algebra they only give some easy examples.

like substitute $\displaystyle u=x^2$ for $\displaystyle x^4+x^2... = u^2+u...$

for factoring. but not how to use with roots and more complicated expressions.
until they show how to do the u-substitution.

8. ## Re: points of intersection

Well, MarkFL2 left out the back substitution.

Basically he finished with the roots u=1 and u=-1.

Since he defined $\displaystyle u=\sqrt[3]x$, back substitution means you still have to solve:
$\displaystyle \sqrt[3]x=1$ and $\displaystyle \sqrt[3]x=-1$.
The solutions for those are x=1 and x=-1.

9. ## Re: points of intersection

Sorry for the omission of the back-substitution...I felt it was evident enough not to mention.

10. ## Re: points of intersection

Originally Posted by ILikeSerena
Well, MarkFL2 left out the back substitution.

Basically he finished with the roots u=1 and u=-1.

Since he defined $\displaystyle u=\sqrt[3]x$, back substitution means you still have to solve:
$\displaystyle \sqrt[3]x=1$ and $\displaystyle \sqrt[3]x=-1$.
The solutions for those are x=1 and x=-1.
As well as \displaystyle \displaystyle \begin{align*} x = e^{-\frac{2\pi i}{3}} , e^{-\frac{\pi i}{3}}, e^{\frac{\pi i}{3}}, e^{\frac{2\pi i}{3}} \end{align*}

11. ## Re: points of intersection

Originally Posted by Prove It
As well as \displaystyle \displaystyle \begin{align*} x = e^{-\frac{2\pi i}{3}} , e^{-\frac{\pi i}{3}}, e^{\frac{\pi i}{3}}, e^{\frac{2\pi i}{3}} \end{align*}
I have to admit that I assumed that the points where the 2 graphs intersect have real coordinates.
Otherwise, we would also need to consider the imaginary solutions for u.

12. ## Re: points of intersection

Originally Posted by ILikeSerena
I have to admit that I assumed that the points where the 2 graphs intersect have real coordinates.
Otherwise, we would also need to consider the imaginary solutions for u.
You're not thinking fourth dimensionally or even third, for that matter :P hahaha.