- #1

basil

- 8

- 0

I have a problem with splitting x

^{4}+ 1 into real quadratic factors. How can this be done?

Cheers.

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In summary, the conversation discusses the process of splitting x^4 + 1 into real quadratic factors. It is suggested to factor the left side and solve the resulting system of equations. However, the speaker also mentions that not all functions may have real quadratic factors and suggests using complex roots to factor in those cases.

- #1

basil

- 8

- 0

I have a problem with splitting x

Cheers.

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- #2

- 22,183

- 3,324

Say we split it as

[tex]x^4+1=(x^2+ax+b)(x^2+cx+d)[/tex]

Try to factor the left side. You'll obtain a system of equations that you need to solve...

- #3

gb7nash

Homework Helper

- 805

- 1

basil said:Hi,

I have a problem with splitting x^{4}+ 1 intoreal quadratic factors. How can this be done?

Cheers.

What do you mean by this? Do you want to factor it into:

x

If this is the case, some functions might not have

x

and multiply. For instance, the only roots of this are the four roots of unity:

r = [tex]e^{\frac{i \pi}{4}} , e^{\frac{i 3 \pi}{4}} , e^{\frac{i 5 \pi}{4}} , e^{\frac{i 7 \pi}{4}}[/tex]

So we can factor into:

x

Now multiply any two arbitary factors together and if you get all real numbers in the quadratic, you have a winner.

--------------

Side note: If you haven't learned about complex numbers yet, I can't think of a better way of doing it than this.

- #4

gb7nash

Homework Helper

- 805

- 1

micromass said:

Say we split it as

[tex]x^4+1=(x^2+ax+b)(x^2+cx+d)[/tex]

Try to factor the left side. You'll obtain a system of equations that you need to solve...

This might be take less time for this problem. I'd race you but I have no paper.

- #5

- 22,183

- 3,324

gb7nash said:If this is the case, some functions might not havereal"quadratic factors".

All real polynomials can be split into real quadratic and linear factors!

- #6

BloodyFrozen

- 353

- 1

Just for reference, and function with a power of 4 is a quartic.

A "Power 4 equation" is a polynomial equation with the highest power being 4. This means that the equation will have terms with variables raised to the power of 4, 3, 2, 1, or 0.

Quadratic factors are expressions that can be factored into two linear expressions. In other words, they are equations in the form of (ax^2 + bx + c) where a, b, and c are constants and x is the variable.

Converting a Power 4 equation to Quadratic factors allows us to solve the equation using methods such as the quadratic formula or factoring. This makes it easier to find the solutions to the equation and understand its behavior.

To convert a Power 4 equation to Quadratic factors, you need to factor out the common term in each term of the equation. Then, you can use the quadratic formula or factoring to solve for the solutions of the equation.

Using Quadratic factors in Power 4 equations allows us to simplify the equation and make it easier to solve. It also helps us to understand the behavior of the equation and make predictions about its solutions.

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