Explanation Theorem used: Rational Zeros Theorem: If the polynomial P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 has integer coefficients provided that a n ≠ 0 and a 0 ≠ 0 , then every rational zero of P is of the form p q , where p and q are integers and p is a factor of the constant coefficient a 0 and q is a factor of the leading coefficient a n . Calculation: The equation is given by x 5 − x 4 − x 3 − 5 x 2 − 12 x − 6 = 0 . By the Rational Zeros Theorem, the rational zeros of P are of the form P = p q , where p is a factor of constant term and q is a factor of leading coefficient. Here, the constant term is − 6 and the leading coefficient is 1. So, P = − 6 1 = − 6 . The factors of − 6 are ± 1 , ± 2 , ± 3 , ± 6 . The possible rational zeros of P are ± 1 1 , ± 2 1 , ± 3 1 , ± 6 1 . To check which of these possible zeros actually are zeros, need to evaluate P at each of these numbers. Use synthetic division, to check whether those values are rational zeros of P . Test whether 1 is a zero as, 1 1 − 1 − 1 − 5 − 12 − 6 1 0 − 1 − 6 − 18 _ 1 0 − 1 − 6 − 18 − 24 From the above calculation, it is observed that the remainder is not 0. So 1 is not a zero. Test whether − 1 is a zero as, − 1 1 − 1 − 1 − 5 − 12 − 6 − 1 2 − 1 6 6 _ 1 − 2 1 − 6 − 6 0 From the above calculation, it is observed that the remainder is 0. So − 1 is a zero. Test whether 2 is a zero as, 2 1 − 1 − 1 − 5 − 12 − 6 2 2 2 − 6 − 36 _ 1 1 1 − 3 − 18 − 42 From the above calculation, it is observed that the remainder is not 0. So 2 is not a zero. Test whether − 2 is a zero as, − 2 1 − 1 − 1 − 5 − 12 − 6 − 2 6 − 10 30 − 36 _ 1 − 3 5 − 15 18 − 42 From the above calculation, it is observed that the remainder is not 0. So − 2 is not a zero. Test whether 3 is a zero as, 3 1 − 1 − 1 − 5 − 12 − 6 3 6 15 30 54 _ 1 2 5 10 18 48 From the above calculation, it is observed that the remainder is not 0. So 3 is not a zero. Test whether − 3 is a zero as, − 3 1 − 1 − 1 − 5 − 12 − 6 − 3 12 − 33 114 − 306 _ 1 − 4 11 − 38 102 − 312 From the above calculation, it is observed that the remainder is not 0

4th Edition

ISBN: 2818000007824

Author: Stewart

Publisher: CENGAGE L

See similar textbooks

Concept explainers

Power Operation

Power operation is topic of algebra in Math. It is use to represent repeated multiplication. Very big number and very small number can be easily express using power operation. Power operation is useful in many fields. In space engineering, it helps in re…

Exponents

The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.

Question

Chapter 3.4, Problem 110E

To determine

To show:

The equation x5−x4−x3−5x2−12x−6=0 has exactly one rational root, and then prove that it must have either two or four irrational roots.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.4, Problem 109E

Chapter 3.5, Problem 1E

Students have asked these similar questions

Schoology X 1. IXL-Write a system of X Project Check #5 | Schx Thomas Edison essay, x Untitled presentation ixl.com/math/algebra-1/write-a-system-of-equations-given-a-graph d.net bookmarks Play Gimkit! - Enter... Imported Imported (1) Thomas Edison Inv... ◄›) What system of equations does the graph show? -8 -6 -4 -2 y 8 LO 6 4 2 -2 -4 -6 -8. 2 4 6 8 Write the equations in slope-intercept form. Simplify any fractions. y = y = = 00 S olo 20

EXERCICE 2: 6.5 points Le plan complexe est rapporté à un repère orthonormé (O, u, v ).Soit [0,[. 1/a. Résoudre dans l'équation (E₁): z2-2z+2 = 0. Ecrire les solutions sous forme exponentielle. I b. En déduire les solutions de l'équation (E2): z6-2 z³ + 2 = 0. 1-2 2/ Résoudre dans C l'équation (E): z² - 2z+1+e2i0 = 0. Ecrire les solutions sous forme exponentielle. 3/ On considère les points A, B et C d'affixes respectives: ZA = 1 + ie 10, zB = 1-ie 10 et zc = 2. a. Déterminer l'ensemble EA décrit par le point A lorsque e varie sur [0, 1. b. Calculer l'affixe du milieu K du segment [AB]. C. Déduire l'ensemble EB décrit par le point B lorsque varie sur [0,¹ [. d. Montrer que OACB est un parallelogramme. e. Donner une mesure de l'angle orienté (OA, OB) puis déterminer pour que OACB soit un carré.

2 Use grouping to factor: 10x + 13x + 3 = 0 Identify A B and C in the chart below feach responce in

Chapter 3 Solutions

ALGEBRA AND TRIGONOMETRY-WEBASSIGN

Ch. 3.1 - To put the quadratic function fx=ax2+bx+c in...Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - 5-8 Graphs of Quadratic Functions The graph of a...Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The equation x 5 − x 4 − x 3 − 5 x 2 − 12 x − 6 = 0 has exactly one rational root, and then prove that it must have either two or four irrational roots.

Solution Summary: The author explains the Rational Zeros Theorem: if the polynomial P(x)=a_n, then every rational zero of P is of the form

Concept explainers

To show:

The equation x5−x4−x3−5x2−12x−6=0 has exactly one rational root, and then prove that it must have either two or four irrational roots.

Want to see the full answer?

Chapter 3 Solutions

The equation x 5 − x 4 − x 3 − 5 x 2 − 12 x − 6 = 0 has exactly one rational root, and then prove that it must have either two or four irrational roots.

Solution Summary: The author explains the Rational Zeros Theorem: if the polynomial P(x)=a_n, then every rational zero of P is of the form

Concept explainers

To show: The equation x5−x4−x3−5x2−12x−6=0 has exactly one rational root, and then prove that it must have either two or four irrational roots.

Want to see the full answer?

Chapter 3 Solutions

To show:

The equation x5−x4−x3−5x2−12x−6=0 has exactly one rational root, and then prove that it must have either two or four irrational roots.