Factoring X^4 + X^2 + (2^0.5)X + 2

Thread starter peripatein
Start date Oct 10, 2012
Tags

Factoring

In summary, when factoring a polynomial with a degree of 4, it is important to first look for common factors and then use factoring techniques such as grouping or difference of squares to simplify the remaining polynomial. The quadratic formula cannot be used to factor polynomials with a degree higher than 2. There are certain patterns and special cases that can make factoring polynomials easier, such as the difference of squares or sum/difference of cubes patterns. Polynomials with a degree of 4 are considered quartic polynomials and can have up to 4 roots or solutions, making factoring more complex and potentially requiring advanced techniques.

Oct 10, 2012

peripatein

X^4 + X^2 + (2^0.5)X + 2 - how may this expression be factorized?

Physics news on Phys.org

Oct 10, 2012

Ray Vickson

Science Advisor

Homework Helper

Dearly Missed

peripatein said:

X^4 + X^2 + (2^0.5)X + 2 - how may this expression be factorized?

What have you tried so far?

RGV

Related to Factoring X^4 + X^2 + (2^0.5)X + 2

1. How do you factor a polynomial with a degree of 4?

To factor a polynomial with a degree of 4, we first need to look for common factors among the terms. In this case, we can see that all terms have a factor of x, so we can factor out an x. This leaves us with x(x^3 + x + (2^0.5) + 2). From here, we can use grouping or other factoring methods to further simplify the remaining polynomial.

2. What is the general process for factoring a polynomial?

The general process for factoring a polynomial is to first look for common factors among the terms, then use factoring techniques such as grouping, difference of squares, or sum/difference of cubes to simplify the remaining polynomial. It may also be helpful to use the rational roots theorem or the quadratic formula to factor polynomials with a degree higher than 2.

3. Can this polynomial be factored using the quadratic formula?

No, the quadratic formula can only be used to factor polynomials with a degree of 2 (quadratic polynomials). This polynomial has a degree of 4, so we cannot use the quadratic formula to factor it.

4. Are there any special cases or patterns to look for when factoring polynomials?

Yes, there are certain patterns or special cases that can make factoring polynomials easier. For example, the difference of squares pattern (a^2 - b^2 = (a + b)(a - b)) or the sum/difference of cubes pattern (a^3 + b^3 = (a + b)(a^2 - ab + b^2) or a^3 - b^3 = (a - b)(a^2 + ab + b^2)). It is also helpful to look for common factors among the terms and to use the rational roots theorem when factoring polynomials with a degree higher than 2.

5. What is the significance of having a polynomial with a degree of 4?

Polynomials with a degree of 4 are considered quartic polynomials. They are one degree higher than cubic polynomials and can have up to 4 roots or solutions. Factoring quartic polynomials can be more complex and may require the use of the rational roots theorem or other advanced factoring techniques.