# Solving Polynomial: How to Expand Square Root

• jessawells
In summary, the conversation discusses how to simplify the expression \sqrt{x^3} + \sqrt{1+x^3} and the group determines that it cannot be simplified further. Some suggestions are made to manipulate the expression, but it is noted that it is already in its simplest form. The conversation also clarifies the definition of a polynomial and how it does not apply to this expression.
jessawells
hi,

can anyone show me how to solve this:

$$\sqrt{x^3} + \sqrt{1+x^3}$$

i want to get it to so that there's only 1 term of x. but i don't know how to expand the squared root. any help is appreciated.

Usually when you say solve an equation, you mean something like solve x + 1 = 2.

Did you mean solve :$$\sqrt{x^3} = \sqrt{1+x^3}$$ (Probably not, since this is a false statement)

or simplify

$$\sqrt{x^3} + \sqrt{1+x^3}$$
?

I'm guessing the second one. Do you have an idea of how to approach it?

Last edited by a moderator:
hi,

yes, its the second case - simplifying the expression. I'm not sure at all how to approach it. I've thought about using the fact that both terms are squared - eg. making it $$\sqrt{x^3 + (1+x^3)}$$ but i know that's wrong. other than that, I've been trying to expand the $$\sqrt{1+x^3}$$ term, the way you would expand something like $$(1+x)^2$$, but I've had no success. any help would be great.

Well Mathematica can't simplify it.

Agreed. That polynomial is in simplest form.

it's not a polynomial, either.

Thank you for giving us the correct definition of a polynomial.

Last edited by a moderator:
No... polynomials only have natural powers. A polynomial is something of the form

$$\alpha_0 + \alpha_1x + \alpha_2x^2 + \ . \ . \ . \ + \alpha_nx^n.$$

which $\sqrt{x^3} + \sqrt{x^3+1}$ certainly is not.

Now, you can certainly do some interesting things to $\sqrt{x^3} + \sqrt{x^3 + 1}$, as usual. For example,

$$\sqrt{x^3} + \sqrt{x^3 + 1} = \frac{1}{\sqrt{x^3+1}-\sqrt{x^3}}$$

$$= \sqrt{2\sqrt{x^3}\left(\sqrt{x^3}+\sqrt{1+x^3}\right) + 1}$$

but I would by no means consider those simpler.

## 1. What is the process for expanding a square root in a polynomial?

The process for expanding a square root in a polynomial involves using the FOIL method, which stands for First, Outer, Inner, Last. This means multiplying the first terms in each set of parentheses, then the outer terms, then the inner terms, and finally the last terms. This will result in a simplified polynomial expression.

## 2. Can you expand a square root without using the FOIL method?

Yes, there are other methods for expanding a square root in a polynomial, such as using the distributive property or factoring out a common term. However, the FOIL method is the most commonly used and efficient method.

## 3. What should I do if I encounter a complex polynomial with multiple square roots?

If you encounter a complex polynomial with multiple square roots, you can simplify it by first expanding each individual square root using the FOIL method. Then, combine like terms and simplify the resulting expression.

## 4. Is it possible to expand a square root with a negative coefficient?

Yes, it is possible to expand a square root with a negative coefficient. When using the FOIL method, you would simply treat the negative sign as you would any other coefficient, and continue with the expansion process.

## 5. How can I check if my expanded square root is correct?

You can check if your expanded square root is correct by simplifying it and comparing it to the original polynomial. If they are equal, then your expansion is correct. You can also use online calculators or ask a math teacher or tutor to check your work.

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