Question

Help me find the intersectio of y=x^(1/3) and y=(x+27)/18

Answer 1

So we can find the points of intersection by setting them equal to one another.

x^(1/3) = (x+27)/18

That much make sense?
From here we'll have to do some tricky steps to solve for x.

Answer 2

Can you please show me the steps I am stuck after trying for one hour to solve this.

Answer 3

Let's start by multiplying each side by 18,
18x^(1/3) = (x+27)

Cubing each side,

(18^3)*x = (x+27)^3

Do you remember how to expand out a 3rd degree binomial like this?
We need to use the 4th row of Pascal's triangle for the coefficients.

Answer 4

So the coefficients will be 1 3 3 1.
And the powers on x will decrease from 3, while the powers on 27 increase from 0 to 3.

(18^3)x = (1)x^3 + (3)x^2(27) + (3)x(27^2) + (1)27^3

We'll get all of our x's on the same side, subtracting the term from the left,

0 = x^3 + 81x^2 - 3645x + 27^3

Answer 5

From here we would have to use the `Rational Root Theorem` to find a root.
The factors of our leading coefficient (1) are +/- 1.
The factors of our constant term (27^3) are +/- 3, 9, 27, 81, 27^2, 27^3, ...

There are a ton more, but let's start with these.
The possible roots will the ratio of the (constant factor) / (leading coefficient factor).

So one possible root is 3/(-1) = -3.
To check it, we have to plug it into the equation and see if it gives us a result of 0 = 0.

This part, unfortunately, is rather tedious.
You have to plug in each combination of factors until you find something that works. :(

Answer 6

To save some time, I'll just tell you that it ends up being that 27 is a root of our polynomial.

We want to divide this root out of our expression.
That will reduce our polynomial to a quadratic which we can finish off using the quadratic formula.

A couple of options are either `Polynomial Long Division` or `Synthetic Division`.
Are you familiar with either method?