# Recent questions tagged polynomials

1 vote
1
A polynomial $p(x)$ is such that $p(0)=5, \: p(1)=4, \: p(2)=9$ and $p(3)=20$. The minimum degree it can have is $1$ $2$ $3$ $4$
2
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 $\mid x_{0} \mid < (n+1)\frac{c_{max}}{\mid c_{1} \mid}.$
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
1 vote
4
The sum of $99^{th}$ power of all the roots of $x^7-1=0$ is equal to $1$ $2$ $-1$ $0$
5
The three roots of the equation $f(x) = 0$ are $x = \{−2, 0, 3\}$. What are the three values of $x$ for which $f(x − 3) = 0?$ $−5, −3, 0$ $−2, 0, 3$ $0, 6, 8$ $1, 3, 6$
1 vote
6
If $x^{2}+ x - 1 = 0$ what is the value of $x^4 + \dfrac{1}{x^4}$? $1$ $5$ $7$ $9$
1 vote
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?
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 $p\left ( -3 \right )<\alpha$ $p\left ( -1 \right )>\alpha$ $p\left ( -1 \right )<\alpha$ $p\left ( -3 \right )<\alpha <p\left ( -1 \right )$
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 $[0,1]$
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 $1$ $0$ $n$ $2$
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
1 vote
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
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 $b=$ _____
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
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 $x^3 +2x^2 +1$ $x^3 +3x^2 -1$ $x^3 +1$ $x^3 -2x^2 +1$
16
If $f(x) = 2x^{7}+3x-5$, which of the following is a factor of $f(x)$? $\left(x^{3}+8\right)$ $(x - 1)$ $(2x - 5)$ $(x + 1)$
17
1 vote
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
1 vote
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 $|\alpha| > 10$.
1 vote
20
The polynomial $x^{4}+7x^{3}-13x^{2}+11x$ has exactly one real root.
21
For any real number $c$, the polynomial $x^{3}+x+c$ has exactly one real root.
1 vote
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
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 $\mathbb{Z}/7\mathbb{Z}[x]$.
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.
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).
1 vote
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?
27
If the cube roots of unity are $1, \omega$ and $\omega^2$, then the roots of the following equation are $(x-1)^3 +8 =0$ $-1, 1 + 2\omega, 1 + 2\omega^2$ $1, 1 - 2\omega, 1 - 2\omega^2$ $-1, 1 - 2\omega, 1 - 2\omega^2$ $-1, 1 + 2\omega, -1 + 2\omega^2$
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 $1$ $2$ $3$ $4$
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? $f(0)f(4)< 0$ $f(0)f(4)> 0$ $f(0)+f(4)> 0$ $f(0)+f(4)< 0$
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 $1$ $2$ $3$ $4$