Solution of "polynomial" with integer and fractional powers

• idmena
In summary, the conversation discusses using a change of variable to solve a polynomial with integer and fractional powers. The suggested approach is to try to eliminate the radical, and a specific example is given.
idmena
Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers.

Homework Statement

I want to solve:
$$x+a(x^2-b)^{1/2}+c=0$$

The Attempt at a Solution

My approach is to make a change of variable x=f(y) to get a true polynomial (integer powers) that I know how to solve, e.g.:
$$y^2+a y +b =0$$
Then I can switch back from y to x and use each of the solutions in y to get solutions in x. I find that works ok when the power of x inside the root is lower or equal than the power of x outside the root, for example, for the following equations:
$$x+a(x-b)^{1/2}+c=0$$
$$x^2+a(x-b)^{1/2}+c=0$$
$$x^2+a(x^2-b)^{1/2}+c=0$$
the solution is given by making the replacement: ##y=(x^n-b)^{1/2}##, then you have a polynomial in y, e.g.: ##y^2+ay+(b+c)=0## which is straightforward.

If I try the same on this one I get a fractional power of y in the new polynomial, which puts me back in square 1. So far I have not been able to find the right change of variable for this problem.

I am trying to work my way up to:
$$a x + b x^2 + c x^3 + d (e+fx+(g+hx+jx^2)^{1/2})^2 =0$$
which is the actual equation I need to solve in the problem I am working on.

Any hints?
Thank you!

The way that I would do it is to try to get rid of the radical. For the simple example:

$x + a\sqrt{x^2 - b} + c = 0 \Rightarrow x+c = - a \sqrt{x^2 - b} \Rightarrow x^2 + 2cx + c^2 = ax^2 - ab$

idmena
stevendaryl said:
The way that I would do it is to try to get rid of the radical. For the simple example:

$x + a\sqrt{x^2 - b} + c = 0 \Rightarrow x+c = - a \sqrt{x^2 - b} \Rightarrow x^2 + 2cx + c^2 = ax^2 - ab$
Of course, it's evident!

1. What is a polynomial?

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division or raising to a power that is not a whole number.

2. What are integer and fractional powers?

Integer powers are when the variable in a polynomial is raised to a whole number, such as x^2 or x^3. Fractional powers, also known as rational powers, are when the variable is raised to a fraction, such as x^(1/2) or x^(3/4).

3. How do you solve a polynomial with integer and fractional powers?

To solve a polynomial with integer and fractional powers, you can use the rules of exponents to simplify the expression and then use algebraic techniques such as factoring or the quadratic formula to find the solutions.

4. Can a polynomial have both integer and fractional powers?

Yes, a polynomial can have both integer and fractional powers. For example, x^2 + 2x^(1/2) - 3 is a polynomial with both integer and fractional powers.

5. Are there any special cases when solving a polynomial with integer and fractional powers?

Yes, there are some special cases to consider when solving a polynomial with integer and fractional powers, such as when the coefficient of the variable is 0 or when the variable is raised to a negative power. These cases may require additional steps in the solving process.

• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
406
• Calculus and Beyond Homework Help
Replies
2
Views
523
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
19
Views
461
• Calculus and Beyond Homework Help
Replies
4
Views
973
• Calculus and Beyond Homework Help
Replies
5
Views
385
• Calculus and Beyond Homework Help
Replies
7
Views
821
• Calculus and Beyond Homework Help
Replies
3
Views
480