Difference of two squares considered to be a quadratic

[vanmaiden]

Homework Statement

is an expression that is a difference of two squares considered to be a quadratic. For example, would x² - 4 be a quadratic? What about x⁴ - 4?

Homework Equations

Ax² + Bx + C

The Attempt at a Solution

I know we can factor a DOTS into two binomials like a quadratic in the for Ax² + Bx + C, but I wanted to be clear on what a DOTS was relative to a quadratic equation.

 

[HallsofIvy]

"Quadratic" simply means "a polynomial of degree 2". Yes, $x^2- 4$ is quadratic, no, $x^4- 4$ is not. The importance of the "difference of two squares" is, as you say, that it can be easily factored: the quadratic $x^2- 4$ can be factored into two linear factors: $(x- 2)(x+ 2)$, the quartic $x^4- 4$ can be factored into two quadratic terms: $(x^2- 2)(x^2+ 2)$.

 

[vanmaiden]

"Quadratic" simply means "a polynomial of degree 2". Yes, $x^2- 4$ is quadratic, no, $x^4- 4$ is not. The importance of the "difference of two squares" is, as you say, that it can be easily factored: the quadratic $x^2- 4$ can be factored into two linear factors: $(x- 2)(x+ 2)$, the quartic $x^4- 4$ can be factored into two quadratic terms: $(x^2- 2)(x^2+ 2)$.

So, to be clear, x⁴ - 4 is a quartic. Is there a special name given to something like x⁶ - 4?

 

[Mark44]

So, to be clear, x⁴ - 4 is a quartic. Is there a special name given to something like x⁶ - 4?

I haven't seen any terminology for polynomials higher than degree five, and these are called quintics.

 

[vanmaiden]

I haven't seen any terminology for polynomials higher than degree five, and these are called quintics.

Interesting. I'll be on the lookout