# Roots of a polynomial and differenciaton

• babita
In summary, if the derivative of a function is zero only once in the interval [a b], then the function can have a maximum of two real roots. This is because there can be cases where the function has only one or no real roots, depending on the graph of the function. For example, a quadratic equation with a negative discriminant will have two complex roots, which are complex conjugates of each other.
babita

## Homework Statement

I read that if f'(x) is zero once in [a b] then f(x) has maximum two real roots.
Why maximum? Shouldn't it be exactly 2?
Or it has something to do with the case of repeated roots?

## The Attempt at a Solution

was thinking as in figure

#### Attachments

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Well in your case it has exactly two real roots, but in another case you might only have one real root or none. Try it out and draw a graph and see what you get. Often there would be one real root and one imaginary root.

oh...like a quadritic equation with D=o , will have f'(x)=0 at -b/2a but it does not have any real roots ?

Thanks :)

Ok take for example x^2+x+1. Does this have any real roots? If yes then what are they?

no it doesn't have any real roots. but f'(x)=0 at x=-0.5
right?

sry typing mistake i there meant D<0

Yes you are correct. You take it's derivative, and you have to see where that derivative equals 0. Like in the equation i gave you 2x+1=0 implies that 2x+1=0 if and only if x=-0.5.

babita said:
sry typing mistake i there meant D<0

Yes if you have the a negative discriminant, then you have two complex roots which are complex conjugates of one another.

## 1. What are the roots of a polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equation equal to zero. They can be found by factoring the polynomial or by using the quadratic formula for higher degree polynomials.

## 2. How do you determine the multiplicity of a root of a polynomial?

The multiplicity of a root is the number of times the root appears in the polynomial equation. It can be determined by factoring the polynomial and looking at the exponents of each factor corresponding to the root.

## 3. What is the relationship between the roots of a polynomial and its graph?

The roots of a polynomial are the x-intercepts of its graph. This means that when the polynomial is graphed, the points where the graph crosses the x-axis are the roots of the polynomial.

## 4. How does differentiation relate to the roots of a polynomial?

Differentiation allows us to find the slope of a polynomial at any given point on its graph. The roots of the polynomial correspond to points where the slope is zero, indicating a horizontal tangent line. This can be helpful in finding the roots of the polynomial.

## 5. Can a polynomial have complex roots?

Yes, a polynomial can have complex roots. Complex roots occur when the polynomial has non-real solutions, which can happen when the polynomial has a degree higher than two. These roots can be found using the quadratic formula or other methods of solving polynomial equations.

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