Find all zeros of the polynomial function or solve the given polynomial equation. Use the Ration

Find all zeros of the polynomial function or solve the given...

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a technique used to estimate the number of positive and negative real roots of a polynomial. By counting the number of sign changes in the sequence of coefficients of the polynomial (and after substituting -x for x to find the negative roots), one can determine an upper bound on the number of positive and negative zeros, which helps in narrowing down the search for the actual roots.

Rational Zero Theorem

The Rational Zero Theorem provides a systematic way to list all possible rational zeros of a polynomial function with integer coefficients. It states that any potential rational zero, expressed in lowest terms, is a fraction whose numerator is a factor of the constant term and whose denominator is a factor of the leading coefficient. This theorem is a vital tool in the factorization and solving of polynomial equations.

Graphical Analysis of Functions

Graphical analysis involves using a graphing utility or plotting the function to obtain a visual representation of the polynomial. This method can provide approximate locations of zeros and insights into the behavior of the graph, such as turning points and end behavior, which assists in confirming the analytical results obtained from other techniques.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The study of polynomial functions includes understanding their degree, the behavior of their graphs, and methods for finding their zeros, which are the values for which the polynomial equals zero.

Zeros of a Polynomial

The zeros of a polynomial are the solutions to the equation where the polynomial is set equal to zero. These zeros can be real or complex and correspond to the x-intercepts of the graph of the polynomial function. Finding these zeros is critical for understanding the function’s behavior and for solving many algebraic and calculus problems.

Transcript

00:02 For this item, you are asked to find the zeros, all of the zeros for the given function.

00:14 So let's start off first by analyzing this a bit.

00:19 Let's list the possible rational roots.

00:23 So in doing so, we're looking at the factors of the final term, which are 1, 2, 5, and 10, the final constant.

00:31 And we would divide those by the factors of the leading coefficient, and that just happens to be one.

00:38 So we have eight options here.

00:41 So let's explore a little bit longer, and let's look at possible root situations.

00:48 So here's a positive.

00:50 So here we've got one.

00:53 From this negative to positive, we've got a second.

00:56 So we could say that there are two or zero possible positive.

01:05 Zeros.

01:09 If we were considering the negative situation, our f of negative x would be x to the fourth plus 4x third minus x squared minus 14x plus 10.

01:30 In here we've got one sign change and here we've got two.

01:35 So it's the same situation.

01:39 It's very possible all the zeros.

01:45 All the zeroes.

01:45 All the zeros could be, we could have all real roots.

01:48 We could have irrational roots.

01:50 We could have two positive and two negative because of course we've got a degree four, a quardic polynomial.

01:59 Or we could have a bunch of imaginaries in there.

02:04 We could have two pairs of imaginaries.

02:07 So when i have a situation like this and the teacher allows me to take a look at the graph, not so much a bad of thing.

02:17 So let's take a peek at the graph and let's pull off one of the real roots.

02:24 If there is what.

02:26 Let's see.

02:27 So let's go to the graph and go with y equals x to the fourth minus 4x cubed minus x squared plus 14 x.

02:48 X.

02:49 Let's double check that equation.

02:50 X to the fourth, 4x cubed, minus x square plus, okay.

02:58 So here we go.

03:00 We find that we have one real rational root, and it's negative one, and it has a duplicity of one.

03:12 A duplicity of two, it's got an even duplicity because it touches the x -axis and bounces off.

03:20 It doesn't cross...

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