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

Polynomial Division

Once a potential zero is confirmed, polynomial division (including synthetic division) is used to factor out the corresponding factor, reducing the degree of the polynomial. This iterative process simplifies the problem by transforming it into one that is easier to solve, ultimately leading to the complete factorization and solution of the polynomial equation.

Graphical Analysis of Polynomials

Using a graphing utility to analyze the behavior of a polynomial function can provide valuable insights into the approximate location of its zeros and the overall shape of the graph. This approach can validate the theoretical predictions made by algebraic methods and assist in identifying initial guesses for iterative methods.

Rational Zero Theorem

This theorem provides a way to generate a list of possible rational zeros of a polynomial with integer coefficients by considering ratios of factors of the constant term to factors of the leading coefficient. It is a crucial tool when trying to narrow down candidates for potential roots before applying more refined techniques, like synthetic division or numerical approximation methods.

Descartes's Rule of Signs

Descartes's Rule of Signs helps to determine the maximum number of positive and negative real zeros in a polynomial function by analyzing the number of sign changes in the sequence of its coefficients. This concept is important for understanding the possible distribution of a polynomial’s roots, guiding further investigation or graphical analysis.

Polynomial Factoring

Factoring is the process of breaking down a polynomial into products of lower-degree polynomials, which makes it easier to solve for its zeros. Recognizing patterns or applying methods like grouping and special factorization formulas can simplify complex polynomials and is a foundational strategy in solving polynomial equations.

Transcript

00:01 For this exercise, we are going to solve the given polynomial equation.

00:08 Let's first begin by listing the, listing the possible rational roots.

00:14 We take the factors of the final constant and divide by the factors of the leading term coefficient, which is just the one.

00:29 So we're working with positive negative 1, 2, 4, and 8.

00:33 We can check descartes ' rule of signs.

00:38 The sign changes, so we have an idea of how many positive negative roots we have or zeros we have.

00:45 Here we've got one sign change, another, and a third.

00:51 So as far as positive roots are concerned, or positive zeros, we have three or one.

01:04 Let's look at the negative case.

01:06 We would have x to the 4th plus x cubed plus 2x squared plus 4x squared plus 4x minus 8 and we would only have this one single sign change so with regard to negative roots or zeros we would only have one so let's use that information and use synthetic division to try to find one, try to find an actual root.

01:47 Let's start with negative 1.

01:51 Let's see how negative 1 works out.

01:55 We've got 1, negative 1, 2, negative 4, negative 8.

02:05 We drop the 1, multiply, we add, we multiply, add, multiply, add, multiply, add, multiply, add, multiply, add, apply and get a zero.

02:22 So we did find zero.

02:27 So one of the zeros here, one of the roots, is x equals negative one.

02:36 Let me separate that.

02:37 It looks like it could be mistaken for a four.

02:41 So let's check.

02:42 Let's fix that.

02:44 X equals negative one.

02:46 That's better.