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Recent questions tagged polynomials

1 vote
1 answer
1
0 votes
0 answers
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Let $c_{1}x^{n} + c_{2}x^{n-1} + \dots + c_{n}x + c_{n+1}$ 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 Lakshman Patel RJIT 19 views
0 votes
1 answer
3
Let $f’(x)=4x^3-3x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to $4x^4-3x^3+2x^2+x+1$ $x^4-x^3+x^2+2x+1$ $x^4-x^3+x^2+2(x+1)$ none of these
asked Sep 18, 2019 in Calculus gatecse 80 views
1 vote
0 answers
4
The sum of $99^{th}$ power of all the roots of $x^7-1=0$ is equal to $1$ $2$ $-1$ $0$
asked Sep 18, 2019 in Quantitative Aptitude gatecse 83 views
5 votes
2 answers
5
The three roots of the equation $f(x) = 0$ are $x = \{−2, 0, 3\}$. What are the three values of $x$ for which $f(x − 3) = 0?$ $−5, −3, 0$ $−2, 0, 3$ $0, 6, 8$ $1, 3, 6$
asked Feb 20, 2018 in Quantitative Aptitude Lakshman Patel RJIT 614 views
1 vote
2 answers
6
If $x^{2}+ x - 1 = 0$ what is the value of $x^4 + \dfrac{1}{x^4}$? $1$ $5$ $7$ $9$
asked Feb 20, 2018 in Quantitative Aptitude Lakshman Patel RJIT 483 views
1 vote
0 answers
7
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 books : 4 of A, 3 of B, ... turn would mean no way of doing probability's tricky questions. EDIT: Here's the images of different questions. How do I differentiate between them?
asked Jun 24, 2017 in Combinatory Vasu Srivastava 406 views
2 votes
1 answer
8
$Q10$ 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 $p\left ( -3 \right )<\alpha$ $p\left ( -1 \right )>\alpha$ $p\left ( -1 \right )<\alpha$ $p\left ( -3 \right )<\alpha <p\left ( -1 \right )$
asked Apr 3, 2017 in Set Theory & Algebra Tesla! 185 views
6 votes
2 answers
9
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 $[0,1]$
asked Apr 3, 2017 in Set Theory & Algebra Tesla! 269 views
2 votes
1 answer
10
$Q8$ 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 $1$ $0$ $n$ $2$
asked Apr 3, 2017 in Set Theory & Algebra Tesla! 146 views
5 votes
2 answers
11
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
asked Apr 3, 2017 in Set Theory & Algebra Tesla! 689 views
1 vote
1 answer
12
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
asked Apr 3, 2017 in Set Theory & Algebra Tesla! 122 views
27 votes
5 answers
13
Consider the quadratic equation $x^2-13x+36=0$ with coefficients in a base $b$. The solutions of this equation in the same base $b$ are $x=5$ and $x=6$. Then $b=$ _____
asked Feb 14, 2017 in Set Theory & Algebra khushtak 6.8k views
11 votes
5 answers
14
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
asked Nov 9, 2016 in Set Theory & Algebra makhdoom ghaya 925 views
4 votes
2 answers
15
The cubic polynomial $y(x)$ which takes the following values: $y(0)=1, y(1)=0, y(2)=1$ and $y(3)=10$ is $x^3 +2x^2 +1$ $x^3 +3x^2 -1$ $x^3 +1$ $x^3 -2x^2 +1$
asked Jun 15, 2016 in Numerical Methods jothee 1.1k views
14 votes
2 answers
16
If $f(x) = 2x^{7}+3x-5$, 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 Quantitative Aptitude Sandeep Singh 2k views
0 votes
1 answer
17
1 vote
0 answers
18
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
asked Dec 17, 2015 in Set Theory & Algebra makhdoom ghaya 84 views
1 vote
1 answer
19
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 makhdoom ghaya 143 views
1 vote
2 answers
20
The polynomial $x^{4}+7x^{3}-13x^{2}+11x$ has exactly one real root.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 200 views
2 votes
1 answer
21
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 141 views
1 vote
1 answer
22
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
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 159 views
3 votes
0 answers
23
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 $\mathbb{Z}/7\mathbb{Z}[x]$.
asked Nov 14, 2015 in Set Theory & Algebra Arjun 213 views
2 votes
1 answer
24
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 tuples $C_{p}, C_{q}$ differ is. At most $d$ At most $n - d$ Between $d$ and $n - d$ At least $n - d$ None of the above.
asked Nov 6, 2015 in Quantitative Aptitude makhdoom ghaya 182 views
12 votes
2 answers
25
For the polynomial $p(x)= 8x^{10}-7x^{3}+x-1$ 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).
asked Oct 30, 2015 in Set Theory & Algebra makhdoom ghaya 738 views
1 vote
1 answer
26
How the following fact applies to determinants (I came across it while solving problems): Consider A is a n&times; n matrix, the elements of which are real (or complex) polynomials in x. If r rows of the determinant become identical when x = a, then the ... how logically connected 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 Mahesha999 949 views
15 votes
3 answers
27
If the cube roots of unity are $1, \omega$ and $\omega^2$, then the roots of the following equation are $(x-1)^3 +8 =0$ $-1, 1 + 2\omega, 1 + 2\omega^2$ $1, 1 - 2\omega, 1 - 2\omega^2$ $-1, 1 - 2\omega, 1 - 2\omega^2$ $-1, 1 + 2\omega, -1 + 2\omega^2$
asked Oct 8, 2014 in Set Theory & Algebra Kathleen 1.2k views
22 votes
3 answers
28
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 Kathleen 3.5k views
20 votes
3 answers
29
A non-zero 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 jothee 2.6k views
28 votes
4 answers
30
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 Kathleen 3.1k views
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