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Chapter 7: Q10E (page 371)

Prove the fundamental theorem of algebra for cubic polynomials with real coefficients.

Short Answer

Expert verified

Cubic polynomials with real coefficients is third degree polynomial, has three roots.

Step by step solution

01

The fundamental theorem of algebra

Every zero can be placed in a zero way on one side of the equator and polynomial of a large or equal degree with real or complex coefficient in the other has at least one root which is a real or complex number.

02

Step 2:Cubic polynomials with real coefficients

Letpx=ax3+bx2+cx+d

Wherea鈮�0

We know that a cubic polynomial must have at least one real root, name itx0 so we can write role="math" localid="1659609692250" p(x)=ax-x0qxwhereqisaquadraticpolynomial.

Now, we know that a quadratic polynomial must always have two roots (counting their multiplicities) real or complex. So, we conclude that third degree polynomial, has three roots.

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Most popular questions from this chapter

find an eigenbasis for the given matrice and diagonalize:

A=[111111111]

23: Suppose matrix A is similar to B. What is the relationship between the characteristic polynomials of A and B? What does your answer tell you about the eigenvalues of A and B?

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[010001100]

IfAis a matrix of rank 1, show that any non-zero vector in the image of Ais an eigenvector of A.

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vector

e鈫�=11..1

InRn is an eigenvector of A. What is the corresponding eigenvalue?
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