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Michael Sipser Edition 3 Exercise 3 Question 21 (Page No. 190)
Let $c_{1}x^{n} + c_{2}x^{n1} + \dots + c_{n}x + c_{n+1}$ be a polynomial with a root at $x = x_{0}.$ Let $c_{max}$ be the largest absolute value of a $c_{i}.$ Show that $\mid x_{0} \mid < (n+1)\frac{c_{max}}{\mid c_{1} \mid}.$
asked
Oct 15, 2019
in
Theory of Computation
by
Lakshman Patel RJIT
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michaelsipser
theoryofcomputation
turingmachine
polynomials
proof
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1
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2
ISI2018DCG10
Let $f’(x)=4x^33x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to $4x^43x^3+2x^2+x+1$ $x^4x^3+x^2+2x+1$ $x^4x^3+x^2+2(x+1)$ none of these
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Sep 18, 2019
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Calculus
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gatecse
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isi2018dcg
calculus
differentiation
polynomials
+1
vote
0
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3
ISI2018DCG11
The sum of $99^{th}$ power of all the roots of $x^71=0$ is equal to $1$ $2$ $1$ $0$
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Sep 18, 2019
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Numerical Ability
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gatecse
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isi2018dcg
numericalability
polynomials
roots
+4
votes
2
answers
4
GATE2018 EE: GA3
The three roots of the equation $f(x) = 0$ are $x = \{−2, 0, 3\}$. What are the three values of $x$ for which $f(x − 3) = 0?$ $−5, −3, 0$ $−2, 0, 3$ $0, 6, 8$ $1, 3, 6$
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Feb 20, 2018
in
Numerical Ability
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Lakshman Patel RJIT
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gate2018ee
generalaptitude
numericalability
easy
polynomials
+1
vote
2
answers
5
GATE2018 CH: GA9
If $x^{2}+ x  1 = 0$ what is the value of $x^4 + \dfrac{1}{x^4}$? $1$ $5$ $7$ $9$
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Feb 20, 2018
in
Numerical Ability
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Lakshman Patel RJIT
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gate2018ch
numericalability
easy
polynomials
+1
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6
Combinatorics : Multinomial Coefficients
What's the relationship between combination and polynomial equation? I mean, I am not able to grasp certain points here or let's say connect them into a whole: 1. Take a question where it's asked that we have to arrange 10 ... way of doing probability's tricky questions. EDIT: Here's the images of different questions. How do I differentiate between them?
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Jun 24, 2017
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Combinatory
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Vasu Srivastava
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277
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permutationandcombination
discretemathematics
engineeringmathematics
polynomials
+2
votes
1
answer
7
ISI 2004 MIII
$Q10$ The equation $p\left ( x \right ) = \alpha$ where $p\left ( x \right ) = x^{4}+4x^{3}2x^{2}12x$ has four distinct real root if and only if $p\left ( 3 \right )<\alpha$ $p\left ( 1 \right )>\alpha$ $p\left ( 1 \right )<\alpha$ $p\left ( 3 \right )<\alpha <p\left ( 1 \right )$
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Apr 3, 2017
in
Set Theory & Algebra
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Tesla!
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isi2004
polynomials
+6
votes
2
answers
8
ISI2004MIII
The equation $\frac{1}{3}+\frac{1}{2}s^{2}+\frac{1}{6}s^{3}=s$ has exactly three solution in $[0.1]$ exactly one solution in $[0,1]$ exactly two solution in $[0,1]$ no solution in $[0,1]$
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Apr 3, 2017
in
Set Theory & Algebra
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Tesla!
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isi2004
polynomials
+2
votes
1
answer
9
ISI 2004 MIII
$Q8$ If $\alpha_{1},\alpha_{2},\alpha_{3}, \dots , \alpha_{n}$ be the roots of $x^{n}+1=0$, then $\left ( 1\alpha_{1} \right )\left ( 1\alpha_{2} \right ) \dots \left ( 1\alpha_{n} \right )$ is equal to $1$ $0$ $n$ $2$
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Apr 3, 2017
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Set Theory & Algebra
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Tesla!
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isi2004
polynomials
+5
votes
2
answers
10
ISI2004MIII7
The equation $x^{6}5x^{4}+16x^{2}72x+9=0$ has exactly two distinct real roots exactly three distinct real roots exactly four distinct real roots six different real roots
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Apr 3, 2017
in
Set Theory & Algebra
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Tesla!
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isi2004
polynomials
maximaminima
+1
vote
1
answer
11
ISI 2004 MIII
If the equation $x^{4}+ax^{3}+bx^{2}+cx+1=0$ (where $a,b,c$ are real number) has no real roots and if at least one of the root is of modulus one, then $b=c$ $a=c$ $a=b$ none of this
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Apr 3, 2017
in
Set Theory & Algebra
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Tesla!
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isi2004
polynomials
+24
votes
5
answers
12
GATE2017224
Consider the quadratic equation $x^213x+36=0$ with coefficients in a base $b$. The solutions of this equation in the same base $b$ are $x=5$ and $x=6$. Then $b=$ _____
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Feb 14, 2017
in
Set Theory & Algebra
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khushtak
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gate20172
polynomials
numericalanswers
settheory&algebra
+10
votes
4
answers
13
GATE19871xxii
The equation $7x^{7}+14x^{6}+12x^{5}+3x^{4}+12x^{3}+10x^{2}+5x+7=0$ has All complex roots At least one real root Four pairs of imaginary roots None of the above
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Nov 9, 2016
in
Set Theory & Algebra
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makhdoom ghaya
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gate1987
polynomials
+4
votes
2
answers
14
ISRO200948
The cubic polynomial $y(x)$ which takes the following values: $y(0)=1, y(1)=0, y(2)=1$ and $y(3)=10$ is $x^3 +2x^2 +1$ $x^3 +3x^2 1$ $x^3 +1$ $x^3 2x^2 +1$
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Jun 15, 2016
in
Numerical Methods
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jothee
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isro2009
polynomials
+13
votes
2
answers
15
GATE20161GA09
If $f(x) = 2x^{7}+3x5$, which of the following is a factor of $f(x)$? $\left(x^{3}+8\right)$ $(x  1)$ $(2x  5)$ $(x + 1)$
asked
Feb 12, 2016
in
Numerical Ability
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Sandeep Singh
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gate20161
numericalability
polynomials
normal
0
votes
1
answer
16
Mock Test 2016 GA
asked
Jan 14, 2016
in
Numerical Ability
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bahirNaik
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testseries
numericalability
polynomials
+1
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0
answers
17
TIFR2014MathsB6
The number of irreducible polynomials of the form $x^{2}+ax+b$, with $a, b$ in the field $\mathbb{F}_{7}$ of $7$ elements is: 7 21 35 49.
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Dec 17, 2015
in
Set Theory & Algebra
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makhdoom ghaya
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tifrmaths2014
polynomials
nongate
+1
vote
1
answer
18
TIFR2011MathsB9
Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $\alpha$ with $\alpha > 10$.
asked
Dec 10, 2015
in
Set Theory & Algebra
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makhdoom ghaya
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tifrmaths2011
polynomials
+1
vote
2
answers
19
TIFR2011MathsA16
The polynomial $x^{4}+7x^{3}13x^{2}+11x$ has exactly one real root.
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Dec 9, 2015
in
Set Theory & Algebra
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makhdoom ghaya
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tifrmaths2011
polynomials
+1
vote
1
answer
20
TIFR2011MathsA11
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.
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Dec 9, 2015
in
Set Theory & Algebra
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makhdoom ghaya
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tifrmaths2011
polynomials
+1
vote
1
answer
21
TIFR2011MathsA8
The sum of the squares of the roots of the cubic equation $x^{3}4x^{2}+6x+1$ is 0. 4. 16. none of the above.
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Dec 9, 2015
in
Set Theory & Algebra
by
makhdoom ghaya
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tifrmaths2011
polynomials
+3
votes
0
answers
22
TIFR2010MathsB15
Which of the following statements is false? The polynomial $x^{2}+x+1$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[x]$. The polynomial $x^{2}2$ is irreducible in $\mathbb{Q}[x]$. The polynomial $x^{2}+1$ is reducible in $\mathbb{Z}/5\mathbb{Z}[x]$. The polynomial $x^{2}+1$ is reducible in $\mathbb{Z}/7\mathbb{Z}[x]$.
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Nov 14, 2015
in
Set Theory & Algebra
by
Arjun
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tifrmaths2010
polynomials
+2
votes
1
answer
23
TIFR2013B2
Consider polynomials in a single variable $x$ of degree $d$. Suppose $d < n/2$. For such a polynomial $p(x)$, let $C_{p}$ denote the $n$tuple $(P\left ( i \right ))_{1 \leq i \leq n}$. For any two such distinct polynomials $p, q,$ the number of coordinates where the ... $d$ At most $n  d$ Between $d$ and $n  d$ At least $n  d$ None of the above.
asked
Nov 6, 2015
in
Numerical Ability
by
makhdoom ghaya
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tifr2013
polynomials
nongate
+8
votes
2
answers
24
TIFR2012A12
For the polynomial $p(x)= 8x^{10}7x^{3}+x1$ consider the following statements (which may be true or false) It has a root between $[0, 1].$ It has a root between $[0, 1].$ It has no roots outside $(1, 1).$ Which of the above statements are true? Only (i). Only (i) and (ii). Only (i) and (iii). Only (ii) and (iii). All of (i), (ii) and (iii).
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Oct 30, 2015
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Set Theory & Algebra
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makhdoom ghaya
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tifr2012
settheory&algebra
polynomials
+1
vote
1
answer
25
Factor of determinant with identical row
How the following fact applies to determinants (I came across it while solving problems): Consider A is a n× n matrix, the elements of which are real (or complex) polynomials in x. If r rows of the determinant become identical when x ... is collapsing of rows of matrix (into one row) with order of its factors. Am I missing some stupid fact here?
asked
Dec 3, 2014
in
Linear Algebra
by
Mahesha999
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417
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757
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matrices
linearalgebra
polynomials
+13
votes
2
answers
26
GATE19952.8
If the cube roots of unity are $1, \omega$ and $\omega^2$, then the roots of the following equation are $(x1)^3 +8 =0$ $1, 1 + 2\omega, 1 + 2\omega^2$ $1, 1  2\omega, 1  2\omega^2$ $1, 1  2\omega, 1  2\omega^2$ $1, 1 + 2\omega, 1 + 2\omega^2$
asked
Oct 8, 2014
in
Set Theory & Algebra
by
Kathleen
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gate1995
settheory&algebra
normal
polynomials
+20
votes
3
answers
27
GATE19974.4
A polynomial $p(x)$ is such that $p(0) = 5, p(1) = 4, p(2) = 9$ and $p(3) = 20$. The minimum degree it should have is $1$ $2$ $3$ $4$
asked
Sep 29, 2014
in
Set Theory & Algebra
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Kathleen
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gate1997
settheory&algebra
normal
polynomials
+18
votes
3
answers
28
GATE201425
A nonzero polynomial $f(x)$ of degree 3 has roots at $x=1$, $x=2$ and $x=3$. Which one of the following must be TRUE? $f(0)f(4)< 0$ $f(0)f(4)> 0$ $f(0)+f(4)> 0$ $f(0)+f(4)< 0$
asked
Sep 28, 2014
in
Set Theory & Algebra
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jothee
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gate20142
settheory&algebra
polynomials
normal
+26
votes
4
answers
29
GATE20002.4
A polynomial $p(x)$ satisfies the following: $p(1) = p(3) = p(5) = 1$ $p(2) = p(4) = 1$ The minimum degree of such a polynomial is $1$ $2$ $3$ $4$
asked
Sep 14, 2014
in
Set Theory & Algebra
by
Kathleen
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52.2k
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gate2000
settheory&algebra
normal
polynomials
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