Use Descartes's Rule of signs to determine the possible...

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a technique used in algebra to determine the maximum possible number of positive or negative real roots of a polynomial equation. It works by counting the number of sign changes in the sequence of coefficients of the polynomial, providing an upper bound on the number of positive roots, while replacing x with -x gives the count for negative roots. This rule aids in understanding the potential real solutions without computing them explicitly.

Real Zeros of Polynomials

The real zeros of a polynomial are the x-values where the polynomial evaluates to zero. Understanding these zeros is crucial as they are the solutions of the equation, and they offer insight into the behavior of the polynomial function, including intervals where the function is positive or negative. Knowledge of the possible number of real zeros helps in further analysis and in determining the nature (real vs. complex) and multiplicity of the roots.

Graphical Verification of Solutions

Graphical verification involves using graphing tools to visually inspect the behavior of a polynomial function and to confirm the analytical findings. By plotting the polynomial, one can observe the points where the graph crosses or touches the x-axis, which represent the real zeros, thereby serving as a practical method to validate results obtained from algebraic rules like Descartes’s Rule of Signs.

Transcript

00:01 This question asks you to use descartes's rule of signs to determine the possible number of positive and negative real zeros for a given function.

00:10 That given function is f of x equals x to the fifth minus x to the fourth plus x cubed minus x squared plus x minus eight.

00:35 So let's go ahead and take a look or analyze f of x to see how many sign changes there are.

00:44 Of course, this is positive.

00:46 So we have one, two, three, four, five.

00:55 Okay.

00:57 So let's record that.

00:59 What that means is that if we are comparing the number of possible and negative real roots, it's possible that we have let's say these are our possible positives possible negatives and possible complex or imaginary we have a possible five positive real zeros we'll continue to build that chart in a minute well we've got a possible five we can do this part a possible five possible three or a possible one i didn't mean to write a two and we can complete this chart in a couple of minutes.

01:44 Let's now inspect f of negative x, substituting in a negative x, wherever we see x, and then going ahead and simplifying all of this.

02:24 I just want to double check that i did this correctly.

02:26 So i had an odd and it changed, even it stayed the same.

02:30 The odd changed.

02:31 The even stayed the same.

02:33 Odd change, evens stayed the same.

02:34 So now when we go through when we are looking for sign changes, there are none.

02:40 So that means that there are zero possible negative real zeros.

02:48 So if that is the case, then if we did have five real positive or positive real zeros, we would have zero negative and zero imaginary.

03:00 If we had three, we would have two imaginary roots or zeros...