Basics of quintic and quadratic expressions

[runicle]

<b>I need a brain refresher</b> to see if i have everything straight in quintic and quadratic expressions.

The n's in an expression represents how many turns a line would have
The amount of (x+1)^2 means a quadratic curve
The if the n is odd there are no complex roots

Now here is the problem... my textbook doesn't explain in full detail which kind of expressions have how many discrete, complex and equal real roots.

 

[turdferguson]

It helps to visualize these functions. (x^2+1) has no roots over reals, while (x^2-1) has 2 real roots. The degree of the polynomial (Im guessing this is what you mean by n) shows how many potential real roots roots it will have. No matter what, the number of real and imaginary roots will equal n.

The degree can also show how many potential changes in direction there will be. For a parabola (maximum of 2 roots), there can be a maximum of 1 change in direction. These are case specific, though, because x^4 looks exactly like a parabola and only has 1 change in direction. A polynomial of 4th can have a maximum of 4 roots and 3 changes in direction. Compare x^4 with (x+1)x(x-1)(x-2) and (x-2)^4.

This should really be moved to the precalc section

 

[m2003]

First of all, don't worry if you need a brain refresher on quintic and quadratic expressions. These concepts can be complex and it's always good to review and make sure you have a solid understanding.

A quadratic expression is an algebraic expression of the form ax^2 + bx + c, where a, b, and c are constants and x is the variable. In other words, it is an expression with a variable raised to the second power. This expression represents a quadratic curve, which is a curve that can be represented by a parabola on a graph. The "n" in an expression does not represent the number of turns, but rather the degree of the polynomial. A quadratic expression has a degree of 2, meaning it has two terms with the variable raised to different powers (x^2 and x^1).

On the other hand, a quintic expression is an algebraic expression of the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f, where a, b, c, d, e, and f are constants and x is the variable. This expression represents a quintic curve, which is a curve that can be represented by a graph with five turning points. The "n" in this expression represents the degree of the polynomial, which in this case is 5.

When it comes to roots, a quadratic expression can have either two distinct real roots, one double root (when the discriminant is equal to 0), or two complex roots (when the discriminant is less than 0). The discriminant is calculated by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic expression. If the discriminant is positive, there are two distinct real roots. If it is 0, there is one double root. And if it is negative, there are two complex roots.

On the other hand, a quintic expression can have up to five distinct roots, but it is not always possible to determine the number of real and complex roots without solving the expression. This is because the number of real and complex roots depends on the specific values of the coefficients in the expression. For example, a quintic expression with all real coefficients can have either five distinct real roots or one real root and two complex roots. However, if the expression has one or more complex coefficients, it can have anywhere from zero to five real roots.

I hope