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

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$x^{4}-x^{3}+2 x^{2}-4 x-8=0$$

Key Concepts

Rational Zero Theorem

This theorem provides a systematic way to list all possible rational candidates for zeros of a polynomial by taking the factors of the constant term and dividing them by the factors of the leading coefficient. It is particularly useful when attempting to identify potential roots by substitution, which can then be verified, to help factor the polynomial or reduce its degree.

Descartes's Rule of Signs

Descartes's Rule of Signs is a method used to estimate the number of positive and negative real zeros in a polynomial equation by counting the number of sign changes in the sequence of its coefficients. This rule helps to narrow down the search for zeros and provides insights into the possible combinations of real and complex roots.

Polynomial Division

Once a potential zero is identified, polynomial division, such as synthetic division or long division, is employed to factor the polynomial and reduce its degree. This process simplifies the problem by breaking down a higher degree polynomial into lower degree factors, which makes it easier to find all remaining zeros.

Graphical Analysis

Graphical analysis involves using graphing tools to visually approximate the zeros of a polynomial. Plotting the function can provide an initial estimate of where the roots lie, confirm the validity of potential zeros suggested by algebraic methods, and help guide further analytical techniques.

Transcript

00:01 So in this question, we are looking to solve x to the fourth minus x cubed plus 2x squared minus 4x minus 8 equals 0.

00:10 So we're going to start by thinking about our rational zero test.

00:15 And we're thinking about the candidates for possible rational zeros here.

00:21 My possible rational zeros can be found by looking at factors of the constant term, in this case negative 8, over factors.

00:31 Of the leading coefficient, in this case one.

00:36 What are my factors of negative 8? well, they are plus or minus 1, plus or minus 2, plus or minus 4, and plus or minus 8, over factors of 1, which are just plus or minus 1.

00:53 And so the division by plus or minus 1 doesn't change anything.

00:59 My candidates for possible rational zeros this time are plus or minus 1.

01:05 Plus or minus two, plus or minus four, and plus or minus eight.

01:10 Yes? and so let's start with one, and we'll use synthetic division to test these possible rational zeros.

01:20 So let's write down our coefficients.

01:23 They are 1, negative 1, 2, negative 4, and negative 8.

01:31 We'll test 1 to start.

01:34 Okay.

01:36 So testing 1, i'm hoping to get 0.

01:38 At the end, drop down the 1, multiply by 1, add, multiply by 1, add, multiply by 1, add, multiply by 1, add, i didn't get 0, and so that didn't work, x equals 1 is not going to be one of my zeros this time.

02:08 And so, so let's check negative 1.

02:12 Okay.

02:13 Now our multiplier is negative 1.

02:15 1 times negative 1, negative 1, add negative 2.

02:22 Multiply by negative 1, add, multiply by negative 1, add, multiply by negative 1.

02:33 And this time, i am in fact getting 0, yes? what does that tell me? it tells me that negative 1 is a 0 and that x plus 1 is a factor of this quartic polynomial...

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