The document contains 20 multiple choice questions about polynomials. The questions cover topics such as solving polynomial equations, finding the remainder of polynomial division, relationships between the coefficients and roots of polynomials, and interpreting graphs of polynomial functions. The correct answers to each question are provided in a key at the end.
This document contains 20 multiple choice questions related to mathematics:
1. The number of digits in the integer part of 7x is 288, where x is a natural number with 2015 digits.
2. Statements I and II are true - I being that a certain function is strictly increasing, and II that a quadratic equation has one real solution.
3. Statements I, II, and III are true - I and II regarding a decreasing sequence being bounded above and below by 0 and 1, respectively, and III regarding being less than 1.
4. The value of T is a particular fraction involving matrices M and N.
The document contains 20 multiple choice questions about polynomials. Some key details:
- Questions ask about finding roots of polynomials, determining if a number is a root, calculating sums and differences of roots, and performing polynomial division.
- Polynomials involve terms up to degree 6 and have coefficients that are both real numbers and variables.
- Questions provide information about roots or evaluations of polynomials in order to determine properties like the degree of a polynomial or the value of a coefficient.
- The answer key is provided at the end for all 20 questions.
The document contains 20 multiple choice questions about complex numbers. The questions cover topics such as: properties of complex numbers and their operations; solving equations involving complex numbers; finding the arguments and moduli of complex numbers; geometric representations of complex numbers; and relationships between complex numbers and trigonometric functions.
This document contains 20 multiple choice questions regarding mathematics concepts such as sets, arithmetic progressions, matrices, trigonometry, polynomials, and complex numbers. The questions test a range of skills including evaluating statements, solving equations, finding values that maximize or minimize expressions, and determining properties of functions and their solutions.
This document contains 23 multiple choice questions about equations. The questions cover a range of topics including solving linear, quadratic, and absolute value equations; finding the number of real solutions of equations; determining the sum or product of the roots of equations; and identifying properties of the solutions sets of equations.
1) The quadratic function f(x) is defined as f(x) = x^2 + cx. For c = 1 - √3, the vertex of the parabola's graph has an x-coordinate and y-coordinate that sum to zero.
2) Three circles are tangent to each other, with radii a = 1 cm, b = 4 cm, and c = 5 cm. For a = 2 cm and b = 3 cm, c must be 10 cm for the triangle formed by the circle centers to be a right triangle.
3) For the linear system defined, the values of m where the sum of squares of the matrix coefficients equals the sum of the matrix elements squared
This document contains 20 multiple choice questions related to mathematics:
1. The number of digits in the integer part of 7x is 288, where x is a natural number with 2015 digits.
2. Statements I and II are true - I being that a certain function is strictly increasing, and II that a quadratic equation has one real solution.
3. Statements I, II, and III are true - I and II regarding a decreasing sequence being bounded above and below by 0 and 1, respectively, and III regarding being less than 1.
4. The value of T is a particular fraction involving matrices M and N.
The document contains 20 multiple choice questions about polynomials. Some key details:
- Questions ask about finding roots of polynomials, determining if a number is a root, calculating sums and differences of roots, and performing polynomial division.
- Polynomials involve terms up to degree 6 and have coefficients that are both real numbers and variables.
- Questions provide information about roots or evaluations of polynomials in order to determine properties like the degree of a polynomial or the value of a coefficient.
- The answer key is provided at the end for all 20 questions.
The document contains 20 multiple choice questions about complex numbers. The questions cover topics such as: properties of complex numbers and their operations; solving equations involving complex numbers; finding the arguments and moduli of complex numbers; geometric representations of complex numbers; and relationships between complex numbers and trigonometric functions.
This document contains 7 questions regarding topics in mathematics and chemistry from the FUVEST 2015 exam.
1) The first question asks to graph a piecewise defined function and find values where it equals 1/5.
2) The second question asks how many words can be formed from an alphabet of two symbols and the minimum size of N to form 1,000,000 words of size N or less.
3) The third question involves writing a system of linear equations from a chemical equation and finding positive integer solutions.
4) The fourth question expresses trigonometric functions in terms of a variable defining a triangle.
5) The fifth question involves solving inequalities.
6) The sixth question relates lengths
This document contains 20 multiple choice questions about complex numbers. The questions cover topics such as solving complex equations, finding roots of complex polynomials, geometric representations of complex number sets, and manipulating complex expressions. Correct answers are provided for each question.
This document contains a summary of a survey given to cadets at the AFA (Air Force Academy) regarding their participation in various sports. The survey found that:
- 66 cadets play volleyball, with 25 not playing another sport
- 68 cadets play swimming, with 29 not playing another sport
- 70 cadets play athletics, with 26 not playing another sport
- 6 cadets play all three sports
The number of cadets that play at least two of the sports is 59.
The document contains a 14-question multiple choice exam on mathematics topics such as geometry, trigonometry, complex numbers, and functions. The exam includes questions about sets, progressions, matrices, volumes, ellipses, logarithms, and more. The key provided at the end of the document indicates the correct answer for each question is choice C for question 1, B for question 2, the answer is nullified for question 3, A for question 4, and so on, with the least number being choice C for the final question.
The document contains 15 multiple choice questions from an exam in Brazil. The questions cover topics in mathematics including geometry, trigonometry, algebra, and calculus.
The document contains 15 multiple choice questions from an IME 2018 exam. The questions cover topics such as determinants, functions, systems of equations, geometry, complex numbers, polynomials, trigonometry, and number representations in different bases. A key is provided with the correct answer for each question labeled A through E.
A tractor factory established a goal to produce 20,000 tractors by 2025, having produced increasing amounts from 2010 to 2017. Assuming continued growth at the same rate, the goal will be reached and surpassed by 150 tractors. Several word problems are presented involving functions, geometry, trigonometry, and other mathematical concepts. The document provides 20 multiple choice questions with answers for an exam.
This document contains a collection of exercises involving determinants of matrices. It includes 17 multiple choice questions testing various properties of determinants such as: how the determinant of a product of matrices relates to the determinants of the individual matrices; how changing elements of a matrix affects its determinant; finding the determinant or elements of an inverse matrix; and conditions under which a matrix is invertible based on its determinant. The questions cover topics like transposes, inverses, properties of determinants, and solving systems of equations using inverses and determinants.
This document contains 20 multiple choice questions about determinants of matrices. The questions cover topics such as calculating determinants, properties of determinants, and conditions under which determinants are equal to certain values. The document also provides the answers to the questions.
The document contains 20 multiple choice questions from an exam in Brazil (ITA 2018). The questions cover a range of math and geometry topics including: arithmetic and geometric progressions, matrices, polynomials, probability, triangles, circles, trigonometry and complex numbers.
This document contains a test with 16 multiple choice questions covering various topics in mathematics. The questions assess knowledge of functions, geometry, probability, matrices, and data analysis. The correct answers are provided at the end.
The document contains a series of word problems involving linear systems. It begins with 17 multiple choice questions related to determining properties of linear systems, solving systems of equations, and applying systems to word problems. The questions cover topics such as determining the conditions for a unique solution, using systems to model real-world scenarios, and properties of the coefficient matrix.
The document contains 15 multiple choice questions about matrices. Some key details:
- Questions ask about properties of matrices like invertibility, multiplication, and Vandermonde matrices.
- Matrices represent things like pixel colors, employee pay, and encoded messages.
- Operations include finding determinants, inverses, eigenvalues, and using matrices to represent geometric transformations.
- The correct answers are usually specific numeric values or matrix expressions.
This document contains 16 multiple choice questions from an exam on various math and logic topics:
1. The value of an expression involving sums and differences of powers of 2 is between 114 and 117.
2. One statement about graphs of two real functions f and g is correct.
3. For functions f, g, and h to have the composite function h∘g∘f have domain R, a condition on m must be met.
The document then provides the answers to each question.
This document contains 20 multiple choice questions about functions. The questions cover topics such as function graphs, function properties and relationships, function composition, and solving equations involving functions. Correct answers are provided for each question.
The document contains 15 multiple choice questions related to circles (circunferências) in the plane. Some key details assessed include:
1) Equations of lines parallel to a given line that intersect a circle and form equal chord lengths.
2) Finding which values satisfy an equation relating a line intersecting a circle.
3) Analyzing statements about triangles, lines, and circles.
4) Calculating the product of symmetrics and conjugates of complex numbers within a given set.
5) Determining the range of a chord length measured by a line intersecting a circle.
The questions cover topics such as parallel lines, secants, tangents, intersections, radii,
The document contains 20 multiple choice questions from an exam for the Brazilian Naval Academy in 2016. The questions cover topics such as systems of equations, probability, geometry, limits, integrals, and other calculus and math concepts.
The document contains 12 multiple choice questions from an exam (ITA 2019). The questions cover a range of mathematical topics including systems of linear equations, probability, geometry, trigonometry, and complex numbers.
The document contains a 20 question multiple choice exam covering various topics in mathematics and geometry. For each question, there are 5 potential answer choices labeled a-e. The questions cover topics such as functions, complex numbers, limits, integrals, geometry, and probability. At the end, a key is provided indicating the correct answer for each question.
The document contains 15 multiple choice questions from an exam in Brazil (IME 2016). The questions cover topics such as sets, trigonometry, probability, geometry, and polynomials. For each question there are 5 possible answer choices labeled a-e. The questions range in difficulty and topic.
The document contains 20 multiple choice questions about mathematics topics such as functions, geometry, trigonometry, and algebra. The questions cover concepts like parabolas, areas, solid geometry, probability, combinatorics, and equations. The document provides a key with the correct answer for each question listed from A to E.
The document contains 20 multiple choice questions about functions. The questions cover topics such as function graphs, function composition, inverse functions, and function properties. They involve identifying function definitions and equations, analyzing function graphs, determining function values and domains, and assessing the truth of statements about functions.
The document contains 10 questions regarding mathematics. Question 1 asks to rewrite a function in a specific form, find its minimum value, and determine the point(s) where it assumes the minimum value. Question 2 asks for the probability of each face of 3 dice summing to 9 showing an odd number. Question 3 asks for the radius of a sphere inscribed in a pyramid. The remaining questions involve solving various equations related to geometry, complex numbers, and number theory. The document provides the questions in Portuguese and includes the answers in a separate section.
The document contains 16 multiple choice questions about mathematics. The questions cover topics such as functions, geometry, trigonometry, probability, and sets. For each question, the correct answer is provided according to the key at the end.
This document contains 10 math problems from an exam in 2015. The problems cover topics like geometry (finding heights of solids cut from a tetrahedron, determining perimeters and areas of triangles), algebra (solving equations, finding real solutions to inequalities), complex numbers (describing loci satisfying an argument equation), and number theory (finding possible integer roots of polynomials, counting polynomials with distinct integer roots).
The document contains 20 multiple choice questions from an exam for the Naval School in 2017. The questions cover topics such as combinations, functions, limits, integrals, complex numbers, and geometry.
This document contains a 15 question multiple choice test covering topics in mathematics. The questions cover areas such as complex numbers, functions, geometry, limits, and equations. The correct answers are provided at the end for questions 1 through 15.
The document contains 20 multiple choice questions about functions. The questions cover topics such as:
- Analyzing graphs of functions and determining function values
- Finding maximums and minimums of functions
- Determining if functions are injective, surjective or inverse functions
- Calculating areas under graphs of functions
This document contains 10 math problems from an exam:
1) There are 150 functions from a set of 5 elements to a set of 3 elements.
2) The graph of the function f(x) = -|x|/2 is sketched.
3) All geometric progressions with a common ratio of 3 and elements from the set {1,2,...,30} are determined. The maximum value of n in the expression m/n with m,n integers and greatest common divisor of 1 is 4.
The document contains 15 multiple choice questions about functions. The questions cover topics such as exponential decay functions, quadratic functions, maximum and minimum values of functions, and function definitions and properties.
O documento apresenta 17 questões do Exame Nacional do Ensino Médio (ENEM) sobre diversos assuntos como: corrida de regularidade, monitoramento de substâncias no sangue, crescimento populacional de médicos, modelos predador-presa, crescimento exponencial de bactérias, ativação de rádio automotivo por código secreto e frequências de transmissão de aparelhos sem fio. As questões envolvem cálculos, interpretação de gráficos e tabelas e raciocínio sobre probabilidades.
O documento apresenta três questões sobre um teste realizado com um novo modelo de carro. A primeira questão descreve que 50 litros de combustível foram colocados no tanque do carro e ele foi dirigido em uma pista de testes até o combustível acabar. A segunda questão fornece um gráfico que relaciona a quantidade de combustível no tanque com a distância percorrida. A terceira questão pede a expressão algébrica que relaciona essas duas grandezas.
O documento descreve um fabricante que decidiu contratar o plano B de uma empresa de entregas, ao invés do plano A que havia escolhido inicialmente. O plano B tem taxa fixa mensal menor, mas taxa variável maior por quilograma enviado. Com 650kg a serem enviados, o plano B terá custo total menor do que o plano A.
O documento apresenta 16 questões do Exame Nacional do Ensino Médio (ENEM) sobre diversos assuntos como: estatística, física, geometria e probabilidade. As questões envolvem interpretação e análise de gráficos, cálculos, resolução de problemas e relações entre grandezas geométricas.
Este documento apresenta 15 questões sobre diversos assuntos como: salário comissionado, interação predador-presa, doenças relacionadas ao saneamento, depreciação de veículos, probabilidades, geometria espacial e volumes de sólidos geométricos. As questões envolvem interpretação e análise de gráficos, cálculos, raciocínio lógico e resolução de problemas.
O documento apresenta 19 questões do ENEM PPL de 2014 sobre diversos assuntos como física, química e matemática. As questões abordam tópicos como emissão de poluentes em veículos, crescimento bacteriano, probabilidade, geometria espacial e outros.
1) O documento apresenta 15 questões do ENEM PPL de 2013 sobre diversos assuntos como matemática, probabilidade e estatística.
2) As questões envolvem cálculos, interpretação de gráficos e tabelas para analisar problemas relacionados a produção industrial, vendas, financiamentos, jogos de azar e outros.
3) As respostas variam entre letras que indicam o resultado correto de cada questão após realizar os procedimentos matemáticos necessários.
1) O documento apresenta 15 questões do ENEM PPL de 2012 sobre diversos assuntos como probabilidades, estatística, geometria e física.
2) As questões envolvem cálculos e análises de gráficos, tabelas e figuras para responder sobre tópicos como produção de resíduos, vendas de produtos, taxas de abandono escolar, capacidade de lixeiras e propriedades geométricas de figuras.
3) São abordados também conceitos como acomodação ocular, convergência de lentes, á
Este documento contém 18 questões do Exame Nacional do Ensino Médio (ENEM) de 2017 sobre diversos assuntos como geometria, funções, probabilidade e estatística. As questões envolvem cálculos, interpretação de gráficos e tabelas e raciocínio lógico.
O documento relata sobre o Exame Nacional do Ensino Médio (ENEM) de 2009 que foi cancelado e traz 15 questões objetivas sobre diversos assuntos como probabilidade, geometria, estatística e análise combinatória.
O documento apresenta 18 questões do ENEM 2010 sobre diversos assuntos como: planejamento de treinos, estimativa de quantidade de estrelas para um painel, volumes de leite em reservatórios, desperdício de água por torneiras, uso de bicicletas compartilhadas, consumo de sacolas plásticas, escolha de estacionamentos, conta de água, necessidade diária de ferro e zinco por meio de alimentos, escolha de museus a visitar, estatísticas de chutes a gol, probabilidade em teste para detecção de
O documento apresenta 16 questões do Enem 2016 sobre diversos assuntos como matemática, física, probabilidade e estatística. As questões abordam tópicos como cálculo de áreas, sistemas lineares, funções exponenciais e probabilidades.
O documento contém 15 questões do Exame Nacional do Ensino Médio (ENEM) da segunda aplicação de 2014. As questões abordam tópicos como matemática, física, biologia, história e língua portuguesa.
O documento descreve os tipos sanguíneos e os resultados de um teste em 200 pessoas. 100 pessoas tinham o antígeno A, 110 o antígeno B e 20 nenhum. Portanto, o número de pessoas com tipo sanguíneo A é igual a 100.
O documento discute um problema de trânsito no Brasil relacionado ao consumo de bebidas alcoólicas por motoristas. Dados mostram que após mudanças no código de trânsito em 2013, como redução do limite de álcool no sangue e aumento de multas, houve queda no número de acidentes entre 2013 e 2015.
The document provides information about 12 multiple choice questions that appeared on the 2018 Brazilian National High School Exam (ENEM). The questions cover topics such as mathematics, statistics, geometry, probability, and other subjects. Specifically, the document provides the questions, answer options, and sometimes additional context or information needed to solve each question.
O documento apresenta 16 questões do Exame Nacional do Ensino Médio (ENEM) de 2017, cobrindo diversos assuntos como geometria, física, probabilidade e estatística. As questões envolvem interpretação e análise de gráficos, cálculos e resolução de problemas.
O documento descreve um problema de engenharia sobre a construção de uma galeria subterrânea para transporte de água entre uma fonte e um reservatório em uma cidade. Dois projetos são apresentados: um segmento de reta ou uma semicircunferência. Após cálculos, o projeto da semicircunferência levaria menos tempo para ser concluído.
Este documento apresenta 16 questões do Exame Nacional do Ensino Médio (ENEM) de 2015 sobre diversos assuntos como física, matemática, probabilidade e estatística. As questões envolvem cálculos, interpretação de gráficos e tabelas para analisar situações problemas.
[1] Um professor alterou as notas de uma prova usando uma função polinomial para compensar questões difíceis. [2] Uma pessoa recebeu propostas de planos de telefonia e pretende gastar R$30,00. [3] A figura mostra a trajetória de um balanço e a equação que a descreve.
Principles of Roods Approach!!!!!!!.pptxibtesaam huma
Principles of Rood’s Approach
Treatment technique used in physiotherapy for neurological patients which aids them to recover and improve quality of life
Facilitatory techniques
Inhibitory techniques
Join educators from the US and worldwide at this year’s conference, themed “Strategies for Proficiency & Acquisition,” to learn from top experts in world language teaching.
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How to Store Data on the Odoo 17 WebsiteCeline George
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About Astro Pathshala
Astro Pathshala is a renowned astrology institute offering comprehensive astrology courses and personalized astrological consultations for over 20 years. Founded by Gurudev Sunil Vashist ji, Astro Pathshala has been a beacon of knowledge and guidance in the field of Vedic astrology. With a team of experienced astrologers, the institute provides in-depth courses that cover various aspects of astrology, including Nakshatras, planetary influences, and remedies. Whether you are a beginner seeking to learn astrology or someone looking for expert astrological advice, Astro Pathshala is dedicated to helping you navigate life's challenges and unlock your full potential through the ancient wisdom of Vedic astrology.
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2. POLINÔMIOS
1
01. (Efomm 2020) Considere a inequação ( )
7 4 2 2
x x x 1 x 4x 3 x 7x 54 0.
− + − − + − − Seja I o conjunto dos números
inteiros que satisfaz a desigualdade e n a quantidade de elementos de I. Com relação a n, podemos afirmar que
a) n é um número primo.
b) n é divisível por 7.
c) n não divide 53904.
d) n é um quadrado perfeito.
e) n é divisível por 6.
02. (Ita 2020) Seja 4 3 2
p(x) ax bx cx dx e
= + + + + um polinômio com coeficientes reais. Sabendo que:
I. p(x) é divisível por 2
x 4;
−
II. a soma das raízes de p(x) é igual a 1;
III. o produto das raízes de p(x) é igual a 3;
IV.
15
p( 1) ;
4
− = −
então, p(1) é igual a
a)
17
.
2
− b)
19
.
4
− c)
3
.
2
− d)
9
.
4
e)
9
.
2
03. (Espcex 2020) Dividindo-se o polinômio 4 3
P(x) 2x 5x kx 1
= − + − por (x 3)
− e (x 2),
+ os restos são iguais. Neste
caso, o valor de k é igual a
a) 10.
b) 9.
c) 8.
d) 7.
e) 6.
04. (Ime 2020) Um polinômio P(x) de grau maior que 3 quando dividido por x 2, x 3
− − e x 5
− deixa restos 2, 3 e
5, respectivamente. O resto da divisão de P(x) por (x 2)(x 3)(x 5)
− − − é
a) 1
b) x
c) 30
d) x 1
−
e) x 30
−
05. (Ita 2020) Considere o polinômio 3 2
p(x) x mx x 5 n,
= − + + + sendo m, n números reais fixados. Sabe-se que toda
raiz z a bi,
= + com 𝑎, 𝑏 ∈ ℝ, da equação p(z) 0
= satisfaz a igualdade 2
a mb nb 1.
= + − Então, a soma dos quadrados
das raízes de p(z) 0
= é igual a
a) 6.
b) 7.
c) 8.
d) 9.
e) 10.
3. POLINÔMIOS
2
06. (Epcar 2020) Considere os polinômios na variável x :
3 3 2
A(x) x (3m 4m)x 2,
= + − − sendo 𝑚 ∈ ℚ; e
2
B(x) x 2x 1
= − +
Os gráficos de A(x) e B(x) possuem apenas um ponto comum sobre o eixo das abscissas.
É correto afirmar que
a) o produto e a soma das raízes imaginárias de A(x) são números conjugados.
b) os afixos das raízes de A(x) formam um triângulo equilátero.
c) as raízes de A(x) possuem argumentos que NÃO formam uma Progressão Aritmética.
d) todas as raízes de A(x) possuem o mesmo módulo.
07. (Espcex 2020) Se a equação polinomial 2
x 2x 8 0
+ + = tem raízesa e b e a equação 2
x mx n 0
+ + = tem raízes
(a 1)
+ e (b 1),
+ centro m n
+ é igual a
a) 2.
−
b) 1.
−
c) 4.
d) 7.
e) 8.
08. (Espcex 2020) Sabe-se que as raízes da equação 3 2
x 3x 6x k 0
− − + = estão em progressão aritmética. Então
podemos afirmar que o valor de
k
2
é igual a
a)
5
.
2
b) 4.
c)
7
.
2
d) 3.
e)
9
.
2
09. (Acafe 2019) Analise as afirmações a seguir e assinale a alternativa correta.
a) A equação 3 2
x 2x 3 0
+ + = possui pelo menos uma raiz irracional.
b) O resto da divisão de 15 4
p(x) x 3x 2x 3
= − + + por q(x) x 1
= + é 3.
c) Se 3 2
p(x) x 5x ax b
= + + + é divisível por x 1
+ e o quociente dessa divisão é um polinômio com raiz dupla então a
e b são primos entre si.
d) Se 3 2 2
2x A Bx C
,
x 2
x 2x 4x 8 x 4
+
= +
−
− + − +
então A B C 1.
+ − =
4. POLINÔMIOS
3
10. (Espcex 2019) Sabendo que o número complexo i (sendo i a unidade imaginária) é raiz do polinômio
5 4
p(x) x 2x x 2,
= − − + podemos afirmar que p(x) tem
a) duas raízes iguais a i, uma raiz racional e duas raízes irracionais.
b) i e i
− como raízes complexas e três raízes irracionais.
c) uma raiz complexa i e quatro raízes reais.
d) i e i
− como raízes complexas e três raízes inteiras.
e) três raízes simples e uma raiz dupla.
11. (Ita 2019) Seja 3 2
p(x) x ax bx
= + + um polinômio cujas raízes são não negativas e estão em progressão aritmética.
Sabendo que a soma de seus coeficientes é igual a 10, podemos afirmar que a soma das raízes de p(x) é igual a
a) 9.
b) 8.
c) 3.
d)
9
.
2
e) 10.
12. (Ita 2019) Considere as seguintes afirmações:
I. se 1 2
x , x e 3
x são as raízes da equação 3 2
x 2x x 2 0,
− + + = então 1 2 3 2 1 3
y x x , y x x
= = e 3 1 2
y x x
= são as raízes da
equação 3 2
y y 4y 4 0.
− − − =
II. a soma dos cubos de três números inteiros consecutivos é divisível por 9.
III.
3 5 1 5
.
2 2
+ +
=
É(são) VERDADEIRA(S)
a) apenas I
b) apenas II
c) apenas III
d) apenas II e III
e) todas
13. (Ime 2019) Sejam 1 2
x , x e 3
x raízes da equação 3
x ax 16 0.
− − = Sendo a um número real, o valor de
3 3 3
1 2 3
x x x
+ + é igual a
a) 32 a
−
b) 48 2a
−
c) 48
d) 48 2a
+
e) 32 a
+
5. POLINÔMIOS
4
14. (Acafe 2018) Considere as funções f(x) 4
= e 3 2
g(x) x 3x .
= − + Os pontos A e B são as intersecções do gráfico
da função g com o eixo das abscissas. Os pontos G e H são as intersecções dos gráficos das funções f e g. O
quadrilátero de vértices ABGH tem área igual a
a) 6 u.a.
b) 4 u.a.
c) 12 u.a.
d) 18 u.a.
15. (Ime 2017) Sejam x, y e z números complexos que satisfazem ao sistema de equações abaixo:
2 2 2
x y z 7
x y z 25
1 1 1 1
x y z 4
+ + =
+ + =
+ + =
O valor da soma 3 3 3
x y z
+ + é
a) 210
b) 235
c) 250
d) 320
e) 325
16. (Ime 2017) O polinômio 3
P(x) x bx 80x c
= − + − possui três raízes inteiras positivas distintas. Sabe-se que duas
das raízes do polinômio são divisoras de 80 e que o produto dos divisores positivos de c menores do que c é 2
c .
Qual é o valor de b?
a) 11
b) 13
c) 17
d) 23
e) 29
17. (Esc. Naval 2017) Seja 6 5 4 3 2
P(x) x bx cx dx ex fx g
= + + + + + + um polinômio de coeficientes inteiros e que
3
P( 2 3) 0.
+ = O polinômio R(x) é o resto da divisão de P(x) por 3
x 3x 1.
− − Determine a soma dos coeficientes de
R(x) e assinale a opção correta.
a) 51
−
b) 52
−
c) 53
−
d) 54
−
e) 55
−
6. POLINÔMIOS
5
18. (Epcar 2017) Sejam Q(x) e R(x) o quociente e o resto, respectivamente, da divisão do polinômio 3 2
x 6x 9x 3
− + −
pelo polinômio 2
x 5x 6,
− + em que 𝑥 ∈ ℝ. O gráfico que melhor representa a função real definida por
P(x) Q(x) R(x)
= + é
a) b) c) d)
19. (Eear 2017) Considere 3 2
P(x) 2x bx cx,
= + + tal que P(1) 2
= − e P(2) 6.
= Assim, os valores de b e c são,
respectivamente,
a) 1 e 2
b) 1 e 2
−
c) 1
− e 3
d) 1
− e 3
−
20. (Acafe 2017) Seja P(x) um polinômio divisível por (x 2).
− Se dividirmos o polinômio P(x) por 2
(x 2x),
+ obteremos
como quociente o polinômio 2
(x 2)
− e resto igual a R(x). Se R(3) 6,
= então, a soma de todos os coeficientes de P(x)
é igual a
a) 38.
−
b) 41.
−
c) 91.
d) 79.
GABARITO
1 - D 2 - D 3 - B 4 - B 5 - B
6 - C 7 - D 8 - B 9 - A 10 - D
11 - A 12 - E 13 - C 14 - C 15 - B
16 - E 17 - E 18 - A 19 - D 20 - B